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A Coxeter group is a group with a presentationwhere $m_{ii}=1$ and $m_{ij}=m_{ji}\in \{2,3,\dots ,\infty \}$ for $i\ne j$. The pair consisting of $W$ and its distinguished generating reflections is a Coxeter system.

A cyclically reduced sequence has no pinch in an HNN extension either internally or across its cyclic junction. If its stable-letter length is positive, every positive power remains reduced, so Britton's lemma gives infinite order. Repeated cyclic conjugation and pinch removal shows that any finite-order element of an HNN extension is conjugate into its base group.

Let $G$ be cyclic and let $R[G]$ carry the regular permutation action. Invariants are constant coefficient vectors, which are the images of the norm operator. A vector with coefficient sum zero is the difference of a vector and its cyclic translate, by successively solving the coordinate differences. Thus the norm group kernel equals $(\sigma -1)R[G]$. Both cyclic Tate cohomology groups vanish. This applies to $R=\mathbb{Z}$ and $R={\mathbb{Z}}_p$, and to finite direct sums of regular lattices.

A cyclic group is a group that can be generated by repeating a single group element $g$ and its inverse element:For an additive group, the same expression is written $\{kg:k\in \mathbb{Z}\}$: $kg$ means repeated addition, including subtraction when $k$ is negative. A finite cyclic group has a repeating cycle, while an infinite cyclic group is isomorphic to the additive integers. The notation $C_n$ refers to a finite cyclic group of order $n$.

This definition compresses a whole composition rule into one repeatable operation. Once $g$ is chosen, $g^a{g}^b=g^{a+b}$, so working with group elements becomes working with integer exponents. In the finite case those exponents are reduced using modular arithmetic. It also proves that every cyclic group is an abelian group, because $a+b=b+a$. A finite cyclic group of order $n$ has exactly $d$ solutions of $x^d=e$ for every positive integer divisor $d$ of $n$: in exponents, $n\mid dk$ has exactly $d$ solutions modulo $n$. The standard finite and infinite models are described in Group Theory (v4.01) by James S. Milne (section 1.16).

The finite cyclic group $C_4=\mathbb{Z}/4\mathbb{Z}$ and the Klein four-group $V_4=C_2\times C_2$ are both abelian groups of order four. In $C_4$, composition is addition modulo four. In $V_4$, label the elements $00,01,10,11$ and add each coordinate modulo two. For example, $01+10=11$ and $11+11=00$.

The order of a group element is preserved by a group isomorphism: an equation $g^m=e$ holds exactly when its image satisfies the same equation. In $C_4$, both $1$ and $3$ have order four. In $V_4$, each nonidentity element has order two. Hence these groups are not isomorphic, despite having the same order of a finite group. No relabelling can repair this discrepancy.

For a geometric model of $V_4$, take the symmetries of a non-square rectangle: the identity element, reflection in its horizontal axis, reflection in its vertical axis, and rotation through $180^{\circ }$. The two reflections commute, and their composite is the half-turn. Every nonidentity motion is its own inverse element. This makes the two binary switches concrete: choosing whether to apply each reflection specifies all four motions. The case of abelian groups of prime-square order generalizes this distinction from $p=2$ to any prime $p$.

A discrete memoryless channel on the additive group $\mathbb{Z}/3\mathbb{Z}$ maps $x$ to $x$ with probability $1-q$ and to $x+1$ with probability $q$. Every input gives the same conditional entropy $h_2(q)$, whereas the output Shannon entropy is at most ${\log }_{2}3$. A uniform input produces a uniform output since every column of the transition matrix sums to one. Hence the channel capacity is ${\log }_{2}3-h_2(q)$ bits per use. This cyclic-error channel differs from a channel distributing its error probability equally over both other output symbols.

For a prime divisor $p$ of the order of a finite group, rotate the tuples $(g_1,\dots ,g_p)$ whose product is the identity. Rotation preserves that constraint even for a nonabelian group, by conjugating the product. Every nonfixed orbit has $p$ elements, while the fixed tuples correspond to $g^p=1$. There are $|G{|}^{p-1}$ tuples, so the number of fixed tuples is divisible by $p$. Since the identity is one, another solution exists and has order exactly $p$. This supplies an elementary proof of Cauchy theorem for groups.

In a finite additive group, a family of $k$-subsets is a difference family of index one if each nonzero group element occurs exactly once among all their ordered within-block differences. Translating its Steiner blocks by every group element gives a Steiner system with parameters $2$-$(v,k,1)$: the unique representation of a difference determines the unique translated Steiner block through two specified points.

Embed a module $M$ over the group ring $\mathbb{Z}[G]$ in a module $I$ with vanishing positive group cohomology, and write $M^{\prime }=I/M$. The connecting map in the long exact group cohomology sequence of $0\to M\to I\to M^{\prime }\to 0$ is an isomorphism $H^r(G,M^{\prime })\to H^{r+1}(G,M)$ for $r\ge 1$, since the groups involving $I$ on both sides vanish. For a finite group and a module $I$ with vanishing Tate cohomology of a finite group in every degree, the same argument shifts every Tate degree. Repeating this construction transports computations to a convenient resolution.

The symmetry group of a cube splits into its central inversion and its rotational symmetry group of a cube. Inversion has determinant $-1$, the rotation subgroup has determinant $1$, and together they generate the full group with trivial intersection. The rotation subgroup acts faithfully on the four body diagonals and is isomorphic to $S_4$. Consequently the full group is this direct product of groups.

A direct product combines two groups into one while keeping their operations in separate coordinates. Its underlying set is the Cartesian product $G\times H$, and its group operation isThe identity element is $(e_G,e_H)$, and the inverse element of $(g,h)$ is $(g^{-1},h^{-1})$. Associativity follows coordinate by coordinate. For finite groups, the multiplication principle gives $|G\times H|=|G||H|$.

For example, write the cyclic group $C_m$ as the additive group of residue classes modulo $m$. In $C_2\times C_3$,since each coordinate reduces with its own integer modulus. A pair can track two cyclic counters that advance independently. Updating the first coordinate does not alter the second, and vice versa.

This independence has an algebraic form. The embedded factor elements always commute:The coordinate maps $(g,h)\mapsto g$ and $(g,h)\mapsto h$ are group homomorphisms, so one can analyze either component separately. This does not make the full product an abelian group when one of its factors is nonabelian: two arbitrary pairs need not commute.

The construction is useful in both directions. It builds a larger group from familiar pieces, while the internal direct product theorem recognizes when an existing group can be separated into such pieces. A cyclic group can sometimes be split this way; whether two finite cyclic factors combine into one cyclic group is decided by the cyclicity of a product of two finite cyclic groups. Definitions and the basic recognition theorems are developed in Thomas Judson, Abstract Algebra: Theory and Applications, §9.2, Direct Products.

For a group of isometries acting properly discontinuously on the hyperbolic plane, and a point $p$ with trivial point stabilizer, the closed Dirichlet domain isIt is the intersection of the closed half-planes bounded by perpendicular bisectors between $p$ and its other group orbit points. The group orbit is locally finite and the hyperbolic plane is a proper metric space, so every point has a nearest group orbit point. This shows that the translates of $D_G(p)$ cover the plane. Distinct interiors are disjoint, since being strictly closer to two different group orbit points is impossible. Boundary ties are allowed in the closed domain. Thus it is a geometric fundamental region.