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Up to conjugation, the proper maximal subgroups of $S_n$ containing a three-cycle are $A_n$; the unequal-part subset stabilizers $S_k\times S_{n-k}$ for $1\le k<n/2$ with $n-k\ge 3$; and the full block stabilizers $S_a\wr S_b$ for $n=ab$, $a\ge 3$, $b\ge 2$. The primitive three-cycle criterion treats the primitive case. A nontrivial block system gives the full permutation wreath product in the imprimitive case. These block stabilizers are maximal because any extra permutation produces a transposition connecting two different blocks; conjugating by the block stabilizer then gives every cross-block transposition, hence the symmetric group. Unequal-part subset stabilizers are maximal by the same connected-transposition argument. Equal-part subset stabilizers are not maximal: they lie in the larger stabilizer that can interchange the two parts.

First prove the primitive three-cycle criterion directly. The useful generation fact is that adding a three-cycle whose support meets a set $S$, with $|S|\ge 3$, to the alternating group $A_S$ generates the alternating group on the enlarged set. If the support is $\{a,b,x\}$ with $a,b\in S$ and $x\notin S$, conjugation by $A_S$ and taking inverses produces every three-cycle with $x$ and two points of $S$. For $|S|\ge 4$ this uses two-transitivity of $A_S$; for $|S|=3$, its cyclic action on pairs together with inversion gives both pair orientations. These cycles, together with the cycles supported inside $S$, are all the three-cycles on $S\cup \{x\}$ and generate $A_{S\cup \{x\}}$. Every even permutation is a product of three-cycles: pair its transpositions, and use $(ab)(bc)=(abc)$ and $(ab)(cd)=(acb)(acd)$, with rightmost factors acting first.

If instead the new support is $\{a,x,y\}$ with just $a\in S$, choose $b\in S\setminus \{a\}$ and conjugate $(axy)$ by an element of $A_S$ carrying $a$ to $b$. This gives $(bxy)$, andThe two-point-overlap case first adjoins $x$; the original cycle then adjoins $y$. This proves connected triple supports generate an alternating group by successively adjoining intersecting supports.

Now take the supports of every conjugate in $G$ of a given three-cycle, and make these triples the edges of a hypergraph on the permutation domain. Its connected components are permuted by $G$, so they form a block system. Since at least one component contains three points, primitivity forces the single component to be the entire domain. Start with one triple and adjoin intersecting triples until all points have been reached. The generation argument shows that the subgroup generated by the conjugate cycles is $A_n$. Hence

To classify the proper maximal subgroups containing a three-cycle, distinguish the three possible types of action. A primitive maximal subgroup contains $A_n$, and thus equals $A_n$, since $[S_n:A_n]=2$. An intransitive maximal subgroup lies in the full stabilizer $S_k\times S_{n-k}$ of one of its orbits, and maximality makes it that full stabilizer. When $k=n-k$, this stabilizer lies properly inside the subgroup that can interchange the two parts, so it is not maximal. Take the smaller part to obtain $1\le k<n/2$. Such an unequal-part stabilizer is maximal: its transpositions connect all points within each part, and any element outside it carries at least one within-part transposition to a cross-part transposition. The resulting connected graph of transpositions generates $S_n$. To see that last assertion, transpositions along a path generate the transposition of its endpoints by successive conjugations. The stabilizer contains a three-cycle exactly when $n-k\ge 3$.

A transitive imprimitive maximal subgroup preserves a nontrivial partition into $b$ blocks of size $a$, with $n=ab$. It equals the full permutation wreath product $S_a\wr S_b$. This full block stabilizer is maximal: any permutation not preserving the partition sends some same-block pair to a different-block pair, giving a cross-block transposition by conjugation. The block stabilizer is transitive on cross-block pairs, so all cross-block transpositions follow; together with its within-block transpositions they generate $S_n$. A three-cycle in this group must act trivially on the blocks, since any nontrivial block permutation moves at least two entire blocks, and a block three-cycle moves $3a$ points. Thus a single three-cycle exists exactly when $a\ge 3$. The full classification, up to conjugation, is thereforeThis is the maximal symmetric-group subgroups containing a three-cycle classification. For $n<3$ there are no such subgroups.

For the first family of counterexamples to extending the criterion to longer cycles, let $G=PS{L}_2(p)$ act on the projective line ${\mathbb{F}}_p\cup \{\infty \}$. The determinant-one matricesact respectively as $x\mapsto x+1$ and $x\mapsto -1/x$. The first fixes infinity and is one $p$-cycle on the finite points. Its translations make the infinity stabilizer transitive on the other points; the second moves infinity, so the full action is two-transitive, hence primitive. The action is faithful after quotienting by scalar matrices. Counting a nonzero first column and the $p$ second columns with determinant one gives $|S{L}_2(p)|=p(p^2-1)$, and the scalar group center has order two. Thussince the ratio on the right to the left is $(p-2)!>1$. Consequently $PS{L}_2(p)$ is the required example for every prime $p>3$.

For the final example take $p=59$, and let the direct product of groups $A_5\times A_5$ act on the sixty elements of $A_5$ byThe action is transitive and the stabilizer of the identity is the diagonal subgroup $\Delta =\{(z,z):z\in A_5\}$. To prove that the diagonal is a maximal subgroup, no classification theorem is needed. If $\Delta <K\le A_5\times A_5$, choose $(x,y)\in K\setminus \Delta$. Multiplying by $(y^{-1},y^{-1})$ gives $(x{y}^{-1},1)\in K$ with $x{y}^{-1}\ne 1$. The subgroup $L=\{z:(z,1)\in K\}$ is normal in $A_5$, by diagonal conjugation, and is nontrivial. Simplicity of $A_5$ gives $L=A_5$, and then $K=A_5\times A_5$. Thus the coset action is primitive. It is also faithful: if $xt{y}^{-1}=t$ for every $t$, then $x=y$ by taking $t=1$, and $x$ is central in the centerless simple group $A_5$, so $x=y=1$. Finally,The prime $59$ is odd, and its primality follows by testing the primes $2,3,5,7$, the only ones below its square root. This is a faithful diagonal primitive permutation group whose order is not divisible by the degree minus one.

The Frattini subgroup is the intersection of the maximal subgroups of $G$, with $\Phi (1)=1$. A useful equivalent property isIndeed, a proper subgroup $H$ lies in a maximal subgroup, which also contains $\Phi (G)$. The Frattini subgroup is a characteristic subgroup, since automorphisms permute the maximal subgroups.

The Fitting subgroup is the largest nilpotent normal subgroup. To prove its existence, first form the p-core $O_p(G)$: the product of all normal p-subgroups. For two such subgroups, their product is normal and has order $|AB|=|A||B|/|A\cap B|$, a power of $p$. There are only finitely many subgroups, so $O_p(G)$ exists. For distinct primes $p,r$,Thus the product of the p-cores is a nilpotent normal subgroup, a direct product of groups of its Sylow subgroups. Conversely, a finite nilpotent normal subgroup $K$ has unique Sylow subgroups, each a characteristic subgroup of $K$. Each is normal in $G$ and lies in the corresponding p-core. Consequently the required largest subgroup exists, and

Choose a minimal nontrivial normal subgroup $L$ of $K$ itself. Every conjugate $L^g$ for $g\in G$ is also a minimal normal subgroup of $K$, since $K$ is normal in $G$. Distinct such subgroups intersect trivially: their intersection is normal in $K$ and minimality forces either equality or trivial intersection. Moreover $[L_i,L_j]\subseteq L_i\cap L_j=1$, so distinct ones commute.

Choose a maximal family of these conjugates whose product $M=L_1\times \cdots \times L_r$ is direct. For any further conjugate $L^g$, its intersection with $M$ is normal in $K$ and is either all of $L^g$ or trivial. In the latter case it commutes with all the existing factors and could be adjoined to the direct product of groups, contrary to maximality. Thus every conjugate is contained in $M$. The product of all conjugates is consequently $M$, which is nontrivial and normal in $G$. Minimality of $K$ as a $G$-normal subgroup gives $M=K$.

Any normal subgroup of a factor $L_i$ is normal in $K$: the other direct factors commute with it, and $L_i$ normalizes it. Minimality of $L_i$ as a $K$-normal subgroup makes it simple. All factors are isomorphic because they are conjugate to $L$. This proves the direct-product structure of a finite minimal normal subgroup:An abelian simple group is cyclic of prime order, so the abelian case is elementary abelian. Now let $H$ be a maximal proper subgroup of a nonabelian simple group $X$. It is nontrivial, since $X$ has nontrivial proper cyclic subgroups. Choose a minimal normal subgroup $K$ of $H$. Then $H\le N_X(K)$. This normalizer cannot be $X$, since that would make $1<K\le H<X$ normal in $X$. Maximality givesFor each of the two groups in this question, the allowed absence of proper nonabelian simple subgroups makes the factors of such a $K$ cyclic of prime order. Hence it suffices to examine normalizers of nontrivial elementary abelian subgroups.

In $A_5$, the possible primes are two, three and five. The Sylow subgroups for the last two primes are cyclic of orders three and five. Every involution is a double transposition; for $(12)(34)$ the centralizer in $A_5$ isthe Klein four-group fixing point five. Thus an elementary abelian $2$-subgroup is cyclic of order two or a Klein four-group. The normalizer of a cyclic order-two subgroup equals its centralizer and has order four, so is contained in a point stabilizer of order twelve. A Klein four-group's common fixed point is intrinsic, and the full point stabilizer $A_4$ normalizes it; its normalizer is therefore exactly that stabilizer.

An order-three subgroup has support a triple. Its normalizer preserves the triple and the remaining pair, and is $(S_3\times S_2)\cap A_5$, of order six. An order-five subgroup acts regularly on all five points. Label the points by ${\mathbb{F}}_5$, with the subgroup acting by translations. Its normalizer in $S_5$ is the twenty affine maps $x\mapsto ax+b$. The translations are even, whereas multiplication by a generator of ${\mathbb{F}}_5^{\times }$ is a four-cycle and is odd. Exactly half these maps are even, so the normalizer in $A_5$ has order ten.

The only candidates for maximal subgroups are thus the normalizers of orders twelve, ten and six. They are all maximal: none can lie in a larger candidate, except potentially an order-six subgroup in an order-twelve $A_4$. But an index-two subgroup of $A_4$ would contain every order-three element; there are eight such elements, plus the identity, already more than six. Hence that containment is impossible. Point stabilizers are conjugate; order-three and order-five subgroup normalizers each form one class by the Sylow theorems. Therefore the maximal subgroups of A5 form preciselyTheir numbers are five, six and ten respectively, since these proper maximal subgroups are self-normalizing in the simple group.

For $X=G{L}_3(2)$, choosing independent columns gives $|X|=(8-1)(8-2)(8-4)=168$. If $t$ is an involution, write $t=I+N$. Field characteristic two gives $N^2=0$, and $\mathrm{im}N\subseteq \ker N$ in dimension three forces its nonzero rank to be one. Thus $t=I+vf$, with nonzero vector $v$, nonzero linear functional $f$ and $f(v)=0$. This transvection fixes the plane $\ker f$ and has image line $⟨v⟩$. Every flag consisting of an incident line and plane occurs, giving $7\cdot 3=21$ involutions, all conjugate. Its centralizer, and hence the normalizer of its cyclic subgroup, is the flag stabilizer of order $168/21=8$. It lies in both the line and plane stabilizers of order $24$, so cannot be maximal.

For the elementary abelian subgroups in GL3 over F2, consider commuting transvections $I+vf$, $I+wh$. Their rank-one parts satisfyIf $v,w$ are independent, equality forces both sides zero; thus $f,h$ vanish on $⟨v,w⟩$ and are the same nonzero functional. If $v=w$, they have a common image line. Consequently the two types of order-four elementary abelian subgroup areNo extra transvection can commute with all elements of $K_L$: if its image were different, its functional would have to equal each of the two independent functionals occurring in $K_L$. Dually, the same holds for $K_W$. Thus these exhaust the elementary abelian $2$-subgroups of rank at least two. The normalizer of $K_L$ is the line stabilizer and the normalizer of $K_W$ is the plane stabilizer, each of order $168/7=24$. They give different conjugacy classes because their common fixed spaces have dimensions one and two. All line stabilizers are conjugate, as are all plane stabilizers.

For an order-three element, its minimal polynomial is $(X-1)(X^2+X+1)$: it has a fixed line and an invariant complementary irreducible plane. Its normalizer preserves both and is the full $G{L}_2(2)$ on that plane, of order six, since the order-three subgroup is normal in $G{L}_2(2)$. This normalizer lies in the corresponding line and plane stabilizers, so is not maximal.

An order-seven element is irreducible. Over the finite field ${\mathbb{F}}_2$ one has $X^7-1=(X-1)(X^3+X+1)(X^3+X^2+1)$, and both cubic factors are irreducible because neither has a root in ${\mathbb{F}}_2$. Its minimal polynomial divides this square-free product and cannot be just $X-1$, since the element is not the identity. It therefore contains a cubic factor; the three-dimensional space forces that irreducible cubic to be the whole minimal polynomial. Identify the space with ${\mathbb{F}}_8$ and the element with multiplication by a generator $\alpha$ of ${\mathbb{F}}_8^{\times }$. Its centralizer consists of the seven nonzero multiplication maps. A normalizer induces an ${\mathbb{F}}_2$-automorphism of this field, so it sends $\alpha$ to $\alpha$, ${\alpha }^2$ or ${\alpha }^4$. All three possibilities occur via the field automorphisms $x\mapsto x^{2^j}$. Therefore the normalizer of the Singer cycle has order $7\cdot 3=21$, and all such normalizers are conjugate by the Sylow theorems.

The maximal-normalizer reduction leaves exactly the two order-twenty-four stabilizer types and the order-twenty-one type. Each is maximal because it cannot be contained in any larger remaining candidate; equal-order subgroups cannot properly contain one another and $21$ does not divide $24$. Thus

The Hall theorem for soluble groups says that for every prime set $\pi$, a finite soluble group has Hall $\pi$-subgroups; they are all conjugate, and every $\pi$-subgroup is contained in one. A Hall $\pi$-subgroup has only primes in $\pi$ in its order and no primes in $\pi$ in its index.

For a proof outline, induct on the group order and choose a minimal normal subgroup $N$. In a finite soluble group this is elementary abelian of some characteristic $p$: its derived subgroup is characteristic and solvability forces it to be trivial; primary components and the subgroup of elements of order dividing $p$ are then characteristic, so minimality gives an elementary abelian p-group. Choose a Hall $\pi$-subgroup of $G/N$ and let $U$ be its preimage. If $p\in \pi$, $U$ is itself a Hall $\pi$-subgroup. If $p\notin \pi$, $U/N$ has order prime to $p$ and coprime splitting over an elementary abelian normal subgroup supplies a complement to $N$ in $U$, of the required Hall order.

Here the complement fact has a short averaging proof. Choose a section of $U/N=H$ and write its multiplication defect as $f(h,k)\in N$, in additive notation. Associativity givesSince $|H|$ is invertible on $N$, averaging over $t$ expresses the defect as $f(h,k)=b(h)+hb(k)-b(hk)$; adjusting the section by $-b$ gives a homomorphic section. Two such complements differ by $d(hk)=d(h)+hd(k)$. Averaging this identity gives $d(h)=b-hb$, so they are conjugate by $b\in N$. Inductive conjugacy in the quotient followed by this complement conjugacy proves Hall conjugacy. To embed an arbitrary $\pi$-subgroup $Q$, first put its quotient inside the chosen quotient Hall subgroup. In the coprime case $Q$ and the corresponding subgroup of a complement are both complements to $N$ inside $NQ$, hence are $N$-conjugate. In the other case $Q\le U$ directly. This proves Hall conjugacy and embedding in finite soluble groups as well as existence.

Now prove the assertion about maximal subgroups by induction on $|G|$. Suppose a nontrivial normal subgroup $K$ is contained in $A$. If $K\nleqq B$, maximality gives $KB=G$, and since $K\le A$ this gives $AB=G$. If $K\le B$, both $A/K$ and $B/K$ are maximal in $G/K$; induction yields either their product equal to $G/K$ or their conjugacy, lifting respectively to $AB=G$ or to conjugacy of $A,B$. The same applies with their roles reversed.

It remains to suppose that no nontrivial normal subgroup of $G$ lies in either maximum. Let $K$ be minimal normal, an elementary abelian p-group. Since $K$ lies in neither maximum, $KA=KB=G$. The intersection $K\cap A$ is normalized by $A$ and by the abelian group $K$, hence by $G$; it is therefore trivial. Likewise $K\cap B=1$. Thus both $A$ and $B$ complement $K$. Their action on $K$ is irreducible, since an invariant subspace would be a proper nontrivial normal subgroup of $G$. It is also faithful: $C_A(K)$ is normalized by $A$ and centralized by $K$, so is a normal subgroup of $G$ contained in $A$ and must be trivial.

If $G/K=1$, these assumptions force $A=B=1$ and $G$ to have prime order. Otherwise choose $L>K$ normal in $G$ with $L/K$ minimal normal in $G/K$. It is an elementary abelian r-group. Under $A\cong G/K$, its copy $R=A\cap L$ is a nontrivial normal subgroup of $A$. The prime $r$ differs from $p$: a p-group acting on the characteristic-p vector space $K$ has nonzero fixed points, by counting its orbits on $K$ modulo $p$. Its fixed subspace would be $A$-invariant, hence all of $K$ by irreducibility, contrary to faithfulness of $R$.

Therefore $A\cap L$ and $B\cap L$ are Sylow r-subgroups of $L=K(A\cap L)=K(B\cap L)$. They are conjugate in $L$. Conjugate $B$ so that these intersections coincide as $R$. Since $R⊲A$, $N_G(R)\ge A$. It cannot equal $G$, since $R$ would then be a nontrivial normal subgroup contained in $A$. Maximality gives $N_G(R)=A$, and the same argument gives $N_G(R)=B$. The conjugated maxima are equal. HenceThis is the factorization-or-conjugacy property of maximal subgroups of a finite soluble group.

A transitive permutation group is primitive when it preserves no nontrivial partition into blocks. A point stabilizer is maximal precisely when the action is primitive. The orbits of any normal subgroup form a block system, so every nontrivial normal subgroup of a faithful primitive group is transitive. This turns the classification of primitive groups into a study of their minimal normal subgroups and their group socle.

The socle of a finite group is generated by its minimal nontrivial normal subgroups. A minimal normal subgroup is characteristically simple, hence a direct power of a simple group. Distinct minimal normal subgroups commute. If a transitive normal subgroup is abelian it is regular: its point stabilizer, being the same at all points, lies in the group homomorphism kernel. In that case it is elementary abelian, and the ambient group is affine. If two minimal normal subgroups occur, each centralizes the other; the centralizer of a transitive permutation group is semiregular, so both are regular and the two regular groups are mutual left/right partners. These observations are central to the O'Nan–Scott theorem.

The theorem organizes finite primitive groups into the following socle/action types. In the affine type HA, the group socle is $C_p^d$, degree $p^d$, and $G=C_p^d⋊H$ with $H\le G{L}_d(p)$ irreducible. In almost-simple type AS the group socle is one nonabelian simple group $T$ and $T\le G\le \mathrm{Aut}(T)$; a maximal subgroup not containing $T$ specifies the primitive coset action.

In simple-diagonal type SD the group socle is $T^r$, $r\ge 2$, and its intersection with a point stabilizer is a full diagonal copy of $T$. The degree is $|T{|}^{r-1}$, and the action includes the appropriate permutation of the simple factors. Compound-diagonal type CD replaces this one diagonal by a product of diagonals in disjoint collections of factors; it is a product version of diagonal type. In product-action type PA one embeds in a wreath product with a primitive nonregular almost-simple base, acting on a Cartesian power ${\Delta }^r$, so the degree is $|\Delta {|}^r$ and the group socle is a corresponding power of $T$.

Types HS and HC have two regular minimal normal subgroups, each isomorphic respectively to $T$ or to $T^r$ with $r>1$. Their socle consists of the left and right regular groups; the degrees are $|T|$ and $|T{|}^r$. Finally TW, twisted-wreath type, has a unique regular nonabelian minimal normal subgroup $T^r$ and a point stabilizer acting through a twisted permutation of its factors. This has degree $|T{|}^r$; primitivity imposes restrictive conditions, in particular $r\ge 6$. These eight modern labels refine the older affine/almost-simple/diagonal/product description. The theorem reduces the problem to simple groups, their maximal subgroups, irreducible linear actions and the specified product constructions; it does not make every primitive group maximal in a symmetric group.

For maximal subgroups of $S_n$ there are first two elementary families. An intransitive maximal subgroup is the stabilizer $S_a\times S_{n-a}$ of an unordered distinguished subset, with $1\le a<n/2$; replacing the subset by its complement gives the same class. When $a=n/2$, this subgroup sits properly inside the stabilizer of the two equal blocks. A transitive imprimitive maximal subgroup is a full block-system stabilizer $S_a\wr S_b$, with $ab=n$, $a,b>1$, in its imprimitive action.

The alternating group $A_n$ is itself maximal. A remaining primitive maximum must contain odd permutations. The primitive part of O'Nan–Scott places it among full affine, diagonal, product-action, or almost-simple overgroups. Full affine groups have degree $p^d$; full diagonal groups have degree $|T{|}^{r-1}$ and socle $T^r$; full product-action wreath groups have degree $d^r$. Almost-simple cases require the actual simple-group permutation degree and its full permutation normalizer. Regular nonabelian types are contained in the corresponding holomorph/diagonal or product overgroups, so do not supply extra maximal families. Parity and exceptional containments must be checked before calling any of these candidates maximal.

In degree sixteen the complete list, up to $S_{16}$-conjugacy, isThe three wreath products here act on blocks, not Cartesian coordinates. The other proper primitive groups of this degree are affine and lie in $AG{L}_4(2)$. A nonzero translation of its sixteen points is eight transpositions, and a generating linear transvection is four transpositions, so the full affine group lies in $A_{16}$. Even the tempting product-action $S_4\wr S_2$ on $4^2$ points is contained in this affine group: identify each four-point coordinate with ${\mathbb{F}}_2^2$. Alternatively, each base transposition repeats four times and coordinate exchange makes six transpositions, both even. Thus none is an additional maximum of $S_{16}$. There are eleven conjugacy classes in the displayed list.

In degree sixty the complete list isThe notation $D(A_5,2)$ denotes the full diagonal normalizer, not just $A_5\times A_5$. It is generated on the set $A_5$ by $x\mapsto ax{b}^{-1}$, an outer automorphism induced by odd conjugation in $S_5$, and $x\mapsto x^{-1}$. Its order isIts two alternating factors form the group socle. The point stabilizer intersects them in the diagonal $A_5$, of index sixty. Odd conjugation by a transposition of $S_5$ fixes its six commuting elements in $A_5$, hence makes $(60-6)/2=27$ transpositions and is odd. Inversion fixes the identity and the fifteen involutions, so makes twenty-two transpositions and is even. This verifies that the full diagonal group is not lost inside $A_{60}$.

The almost-simple maximum $PG{L}_2(59)$ has order $59(59^2-1)=205320$ and its usual two-transitive projective-line action. Multiplication by a primitive element of ${\mathbb{F}}_{59}^{\ast }$ makes one 58-cycle and fixes zero and infinity, so it too contains odd permutations. The simple subgroup $PS{L}_2(59)$ alone is not maximal. Degree sixty is neither a prime power nor a proper power, excluding affine and product/compound types; its diagonal possibility is $T=A_5,r=2$. The almost-simple primitive possibilities, besides the natural alternating/symmetric groups, have socle $PS{L}_2(59)$. Their full normalizer is the displayed $PG{L}_2(59)$. These degree-specific classification facts supply the exhaustive primitive part of the list. Altogether there are forty-two conjugacy classes of maximal subgroups of $S_{60}$: twenty-nine intransitive, ten imprimitive, the alternating group, and two further primitive classes.

For a finite group, define the Frattini subgroup bythe intersection is $G$ when there are no maximal subgroups, which for a finite group happens only for $G=1$. A second definition is the set of non-generators: an element $x$ such that $⟨S,x⟩=G$ always implies $⟨S⟩=G$.

If $x\in \Phi (G)$ and $H=⟨S⟩<G$, put $H$ in a maximal subgroup $M$. Then $x\in M$, so $⟨S,x⟩\le M<G$. Conversely, if $x\notin \Phi (G)$, some maximal subgroup $M$ omits it; maximality gives $⟨M,x⟩=G$, although $M\ne G$. This proves equivalence.

Every group automorphism permutes maximal subgroups, so $\Phi (G)$ is a characteristic subgroup, in particular a normal subgroup. Alsoa proper $H$ would lie in a maximal subgroup containing both factors.

Set $F=\Phi (G)$ and take a Sylow subgroup $P$ of $F$. For $g\in G$, normality of $F$ makes $P^g$ a Sylow subgroup of $F$. Conjugating inside $F$ therefore gives the Frattini argument factorization $G=F{N}_G(P)$. The preceding generating property forces $N_G(P)=G$. Thus every Sylow subgroup of $F$ is normal, and part (i) shows that $F$ is a nilpotent group.

Now suppose $G$ is a finite p-group. The normalizer condition makes every maximal subgroup $M$ normal: $M<N_G(M)$ and maximality force $N_G(M)=G$. The quotient $G/M$ has no proper nontrivial subgroup, hence is cyclic of prime order; its p-power order makes that prime $p$. Consequently every such $M$ containswhere the two factors are the commutator subgroup and the subgroup generated by pth powers. Thus $D\le \Phi (G)$.

The quotient $G/D$ is an elementary abelian group, hence a vector space over ${\mathbb{F}}_p$. If $x\notin D$, extend the nonzero vector $xD$ to a basis and choose a linear functional nonzero on it. Its kernel pulls back to a subgroup of index $p$, a maximal subgroup omitting $x$. Therefore $x\notin \Phi (G)$, provingThe two factors in the last formula are multiplied; the superscripted expression in the TeX aid is not the formula printed in the PDF.