...
order other than the unit element (see [[
Order|
Order]] of an element of a
group).
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.G. Kurosh, "The
theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian
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''dihedron group''
...In a finite group, two different elements of order 2 generate a dihedral group.
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''Hamiltonian group''
...rticular, any Hamiltonian group is periodic (cf. [[Periodic group|Periodic group]]).
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...umber) is metacyclic. Polycyclic groups (cf. [[Polycyclic group|Polycyclic group]]) are a generalization of metacyclic groups.
...the more special class of groups whose derived group and derived quotient group are both cyclic.
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...cept $n=4$, this group is simple; this fact plays an important role in the theory of solvability of algebraic equations by radicals.
Note that $A_5$ is the non-Abelian simple group of smallest possible order.
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''torsion-free group''
...with respect to two different prime numbers $p$, then it is a torsion-free group.
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...rder 8 is the smallest finite group that is not a T-group. A group is a T-group if and only if it is equal to its own [[Wielandt subgroup]].
* Derek Robinson, "A Course in the Theory of Groups", Graduate Texts in Mathematics '''80''' Springer (1996) {{ISBN|0
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...e of subgroups of a group is a [[distributive lattice]] if and only if the group is locally cyclic.
* Marshall Hall jr, ''The Theory of Groups'', reprinted American Mathematical Society (1976)[1959] {{ISBN|0-
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''equi-affine group''
The subgroup of the general [[affine group]] consisting of the affine transformations of the $n$-dimensional affine sp
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...up, proved by L. Sylow [[#References|[1]]] and playing a major role in the theory of finite groups. Sometimes the union of all three theorems is called Sylow
Let $G$ be a finite group of order $p^ms$, where $p$ is a prime number not dividing $s$. Then the following th
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...$ and its centralizer $K$ in $G$; indeed $K$ is isomorphic to the quotient group $G/B$.
...form space]] with respect to the [[uniformity]] implied by the topological group structure.
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$#C+1 = 31 : ~/encyclopedia/old_files/data/P071/P.0701710 Partially ordered group
A [[Group|group]] $ G $
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''cyclic semi-group''
...le$, then $a,\dots,a^{h+d-1}$ are distinct elements and, consequently, the order of $A$ is $h+d-1$; the set
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''soluble group''
...] of a group). The term "solvable group" arose in [[Galois theory|Galois theory]] in connection with the solvability of algebraic equations by radicals.
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...gebras was discovered by C. Chevalley [[#References|[2]]] (cf. [[Chevalley group]]). In particular, Chevalley's method makes it possible to obtain Dickson g
[[Category:Group theory and generalizations]]
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...Every pure linear sub-semi-group $P$ of an arbitrary group defines a right order, namely $x<y$ if and only if $yx^{-1}\in P$.
...ll subgroups in $S(G)$ become convex. In a locally nilpotent right-ordered group the system of convex subgroups is solvable.
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...r groups are said to be cyclic (they are isomorphic to either the additive group $\mathbf Z$ of integers, or the additive groups $\mathbf Z_n$ of residue cl
...groups that are simple (cf. [[Finitely-presented group|Finitely-presented group]]).
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A metabelian $2$-group (cf. [[Meta-Abelian group|Meta-Abelian group]]) of order 8, defined by generators $x,y$ and relations
The quaternion group can be isomorphically imbedded in the multiplicative group of the algebra of quaternions (cf. [[Quaternion|Quaternion]]; the imbedding
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''$p$-component of a group element of finite order''
...or $p$-component of $x$ and $z$ is the $p'$-part or $p'$-component. If the order of $x$ is $r=p^{\alpha}s$, $(p,s)=1$, $bp^{\alpha}+cs=1$, then $y=x^{sc}$,
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A [[P-group| $ p $-
group]] $ G $
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