The same as an inverse semi-group (cf. [[Inversion semi-group|Inversion semi-group]]).
[[Category:Group theory and generalizations]]
147 bytes (18 words) - 17:45, 15 November 2014
''dihedron group''
...In a finite group, two different elements of order 2 generate a dihedral group.
1 KB (202 words) - 16:24, 19 October 2014
...sentation of $G$ on the set of right cosets of $H$ in $G$ (cf [[Coset in a group]]). Then its kernel is the core of $H$ in $G$.
.../TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W.R. Scott, "Group theory" , Dover, reprint (1987) (Original: Prentice-Hall, 1964)</TD></TR></table
802 bytes (136 words) - 21:48, 30 November 2016
...[Group|Group]]), where the [[Frattini-subgroup(2)|Frattini subgroup]] of a group plays an important role. Due to the many close connections which Lie algebr
...and algebras]]) is the intersection of its maximal subalgebras. Unlike the group case, where the Frattini subgroup is always a [[Normal subgroup|normal subg
2 KB (323 words) - 13:52, 25 April 2014
''equi-affine group''
The subgroup of the general [[affine group]] consisting of the affine transformations of the $n$-dimensional affine sp
1 KB (158 words) - 22:38, 2 November 2014
...between the elements of any finite generating set of a finitely-presented
group contains a finite set of defining relations in these generators.
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "The
theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian
2 KB (242 words) - 18:26, 26 October 2014
...perators on $E$. The space $E$ is called the representation space of $\pi$ and the operators $\pi(x)$, $x\in X$, are called the operators of the represent
...valign="top">[1]</TD> <TD valign="top"> A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)</TD></TR>
1 KB (197 words) - 19:18, 25 October 2014
...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]]
1 KB (187 words) - 21:05, 15 November 2017
...of $G$ coincides with the rank of $\widehat G$ (cf. [[Rank of an algebraic group]]).
...">[a2]</TD> <TD valign="top"> V.S. Varadarajan, "Lie groups, Lie algebras, and their representations" , Prentice-Hall (1974) {{MR|0376938}} {{ZBL|0371.220
1 KB (177 words) - 19:00, 7 April 2023
...mi-group]] with unit (i.e. [[Monoid]]) satisfying the [[cancellation law]] and in which any non-invertible element $a$ is decomposable into a product of i
a = b_1 \cdots b_k\ \ \text{and}\ \ a = c_1 \cdots c_l
1 KB (153 words) - 16:17, 21 December 2014
...). As an operation on a class of groups, the verbal product is associative and, within the corresponding [[variety of groups]], it is also free.
...litar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience) (1966) {{ZBL|0138.25604}}. pp. 412</TD><
883 bytes (142 words) - 19:33, 11 December 2015
...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]]).
2 KB (343 words) - 18:24, 26 October 2014
''Lie group of type $(E)$''
A real finite-dimensional [[Lie group|Lie group]] $G$ for which the [[Exponential mapping|exponential mapping]] $\exp\colon
2 KB (253 words) - 22:20, 14 November 2014
''triangular Lie group''
...f. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]) are real for any element $g$.
2 KB (303 words) - 18:20, 12 December 2019
...as an increasing ideal series (see [[Ideal series|Ideal series]] of a semi-group) such that for any two adjacent terms $A_\alpha,A_{\alpha+1}$,
...for right (or left) ideals. If all nilpotent sub-semi-groups of a nil semi-group $S$ are finite, then so is $S$.
2 KB (360 words) - 17:24, 14 October 2014
''of a group $A$ by a group $B$''
...ruct a semi-direct product one should also know which automorphisms of the group $B$ are induced by conjugation by elements of $A$. More precisely, if $G =
2 KB (359 words) - 16:54, 23 November 2023
''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.
3 KB (443 words) - 18:25, 26 October 2014
...of L. Sylow, who proved a number of theorems on such subgroups in a finite group (see [[Sylow theorems|Sylow theorems]]).
...f its Sylow $2$-subgroups. In the theory of infinite groups, except in the theory of locally finite groups, the role of Sylow subgroups is less important, si
2 KB (324 words) - 19:15, 4 April 2023
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...en the derived group of $G$ is locally finite and, in fact, locally normal and Chernikov.
2 KB (350 words) - 16:55, 1 July 2020
...$ with degree prime to $p$ (cf. also [[Character of a group|Character of a group]]). The simplest form of the McKay–Alperin conjectures asserts that
...]]] first suggested this might be true when $G$ is a [[Simple group|simple group]]. J.L. Alperin [[#References|[a1]]] observed that it is probably true for
2 KB (358 words) - 18:45, 13 October 2014