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Put $\overline{G}=G/O_{p^{\prime }}(G)$ using the p-prime core of a finite group. A finite group is p-constrained if $C_{\overline{G}}(O_p(\overline{G}))\le O_p(\overline{G})$. If $O_{p^{\prime }}(G)=1$, the p-core of a finite group has its centralizer contained in itself. Then block idempotents centralize a normal p-subgroup places every block idempotent in $k{O}_p(G)$, a local ring, so $kG$ has only the principal block.

The p-core is the largest normal p-subgroup of a finite group. The product of two normal p-subgroups is again a normal p-subgroup, so this subgroup exists and is characteristic.

For a connected source, first integrate on its universal covering Lie group. The resulting map descends to the original group exactly when it kills the discrete kernel of the covering map. For the circle group, a real-linear map $t\mapsto ct$ integrates to a circle endomorphism exactly when $c\in \mathbb{Z}$, since $2\pi \mathbb{Z}$ must map into itself. This is a global period condition invisible to the zero bracket of the circle's Lie algebra.

If a group $K$ acts on a finite set $X$, it acts on the direct product of groups $H^X$ by permuting its factors. The resulting semidirect product is the permutation wreath product. For $K=S_m$ acting on $m$ two-point blocks and $H=C_2$, it describes independent within-block swaps combined with a permutation of blocks, and has order $2^{m}m!$. The action on $X$ is part of the definition; it need not be the regular action used in a regular wreath product.

Suppose nontrivial groups $G_1,G_2$ act on a set and there are disjoint nonempty subsets $X_1,X_2$ such that every nonidentity element of $G_1$ maps $X_2$ into $X_1$, while every nonidentity element of $G_2$ maps $X_1$ into $X_2$. Assume at least one factor has at least three elements. Then the subgroup generated by $G_1$ and $G_2$ is their free product. The cardinality hypothesis excludes the counterexample in which $G_1=G_2=C_2$ acts by swapping $X_1$ and $X_2$. Versions allowing both factors to have order two need an additional separation hypothesis.

A pro-p group whose closed commutator subgroup lies in its closed power subgroup $G^{p^{ϵ}}$, with $ϵ=1$ for odd $p$ and $ϵ=2$ for $p=2$. This is the profinite analogue of a powerful p-group. Finitely generated torsion-free examples are exactly the uniform pro-p groups. The additive group of a finite product of p-adic integers is an abelian example.

A profinite group $G$ is procyclic if every quotient by an open normal subgroup is cyclic. Equivalently $G=\overline{⟨g⟩}$ for one element $g$. For the forward implication, the sets of lifts of generators of $G/N$ are nonempty clopen sets. Any finite intersection contains a lift of a generator of the cyclic quotient by the intersection of those open normal subgroups. Compactness supplies one element generating every finite quotient, and hence generating densely. The reverse implication follows by projecting that generator. For a pro-p group, two distinct maximal open subgroups would give a quotient $C_p^2$, so a nontrivial procyclic group has a unique maximal open subgroup $M$. Its profinite Frattini subgroup is $M$, and any $g\notin M$ generates topologically: otherwise the proper closed subgroup it generates lies in a maximal open subgroup. The additive group of p-adic integers is an infinite example, generated topologically by $1$.

The inverse limit of the quotients of an abstract group by its normal subgroups of finite $p$-power index. Maps to pro-p groups factor uniquely through the canonical dense image in this completion. Unlike profinite completion, only finite p-groups are used.

A pro-$p$ ring is a topological ring with continuous ring operations whose underlying additive group is a pro-p group. Standard local analytic examples are complete Noetherian local rings with finite residue field of characteristic $p$, such as ${\mathbb{Z}}_p$ and ${\mathbb{F}}_p[[t]]$. The topology is essential to the convergence of power series used for analytic group operations.

The Prüfer rank bounds the number of generators of a group needed for each finitely generated subgroup, without requiring the ambient group to be finitely generated. The additive group $\mathbb{Q}$ has Prüfer rank one: any finitely generated subgroup has a common denominator and is cyclic, but no finite collection generates all rational numbers. Prüfer rank cannot increase under taking subgroups or quotients; for quotients, lift a finite generating set and apply the rank bound to the subgroup generated by its lifts.

In an accepted padded convolution of words, a loop after one tape ends can be deleted or repeated while keeping that entire tape fixed. The altered other tape still represents the same required neighbouring group element. A shortest representative has a suffix no longer than the fixed number of automaton states; an accepted longer suffix gives infinitely many representatives of the same element.

For a group $G$ and a normal subgroup $N$, the quotient group $G/N$ is the set of cosets $gN$, with operationIts identity element is the coset $N$, and the inverse element of $gN$ is $g^{-1}N$. A quotient group treats all elements within one coset as the same input while retaining a consistent operation on the classes. The normal subgroup condition is what makes this operation independent of which representatives are chosen. ^[ref].

Take the additive group $G=\mathbb{Z}/6\mathbb{Z}$ and $N=\{0,3\}$. Since $G$ is an abelian group, $N$ is a normal subgroup. There are three elements in the quotient group:Adding two copies of $2+N$ gives $4+N=1+N$; adding $1+N$ and $2+N$ gives $3+N=N$. Hence the quotient group has the addition table of $\mathbb{Z}/3\mathbb{Z}$. It does not select one representative and discard the others: its elements are the entire cosets.

This is useful whenever distinctions within a normal subgroup should be ignored. For example, the sign of a permutation retains only even or odd parity of a permutation in $S_3$. Its group kernel is the alternating group $A_3$, and the quotient group $S_3/A_3$ has two elements. The first isomorphism theorem explains why a group homomorphism's image is a quotient group of its domain.

The quotient map of groups sends each group element to its coset modulo a normal subgroup. It is a surjective group homomorphism, with group kernel exactly $N$. In the additive six-element example, it sends $0,3$ to $N$, $1,4$ to $1+N$, and $2,5$ to $2+N$. The quotient map of groups records the identification that produces the new group.

A reduced word contains no adjacent generator and inverse-generator pair. Cancel such pairs to obtain the unique reduced representative in a free group. One proof constructs the free group itself from reduced words, with multiplication given by concatenation followed by cancellation; its action by these operations on words distinguishes distinct reduced representatives.

A transitive group action is regular if every point stabilizer is trivial. Equivalently each ordered pair of points is connected by exactly one group element. Identifying one point with the identity identifies the action with left multiplication on the group. A regular normal subgroup inside a larger primitive group can supply translations or nonabelian left multiplication; the whole primitive group usually has a nontrivial stabilizer acting by automorphisms of this regular subgroup.