The [[Exponent of a group|exponent]] of the multiplicative group of integers modulo $n$, denoted $\lambda(n)$; the least positive integer $\
Since the multiplicative group modulo an odd prime power $p^a$ is cyclic, in these cases we have $\lambda(
1 KB (186 words) - 16:57, 25 November 2023
...ad. A Baer group is maximal of elation, respectively homology, type if the group acts transitively on the non-fixed points on each line $L$ of the spread wh
...gn="top">[a2]</TD> <TD valign="top"> V. Jha, N.L. Johnson, "Structure theory for point-Baer and line-Baer collineation groups in affine planes" , ''Proc
2 KB (305 words) - 17:24, 2 September 2017
A norm on a skew-field the group of values of which is isomorphic to the group of integers $ \mathbf Z $.
is also sometimes understood as the norm having as group of values the $ r $-
1 KB (165 words) - 19:36, 5 June 2020
...cally finite group]]), finiteness of rank (cf. [[Rank of a group|Rank of a group]]), and finiteness of conjugacy classes (cf. [[Conjugate elements|Conjugate
...t groups (cf. [[Nilpotent group|Nilpotent group]]) it is equivalent to the group itself being finitely generated.
5 KB (720 words) - 17:15, 7 February 2011
...subgroups are conjugate in $G$. Any $\pi_1$-subgroup of a $\pi$-separable group $G$ is contained in some $\pi_1$-Hall subgroup of $G$ (see [[#References|[2
...and if all prime divisors of $k$ are in $\pi$, then $G$ has a subgroup of order $k$ and all these subgroups are conjugate in $G$. If $\pi$ consists of all
2 KB (379 words) - 09:33, 27 October 2014
...23 : ~/encyclopedia/old_files/data/M064/M.0604790 Monomial substitutions, group of
The subgroup of the group $ \mathop{\rm GL} ( m , \mathbf Z [ H ] ) $
2 KB (302 words) - 08:01, 6 June 2020
...ollowing generalization of Frobenius' theorem: If $G$ is a finite group of order $g$ and $C$ is a [[conjugacy class]] of $G$ of cardinality $h$, then the nu
...mple, $x^4=1$ has exactly $4$ solutions in the [[Symmetric group|symmetric group]] on three letters, but obviously the solutions do not form a subgroup of $
4 KB (650 words) - 20:59, 29 November 2014
...on, a [[Totally ordered set|totally ordered set]] with respect to a binary order relation $\leq$ and satisfies the following axiom: For any elements $x,y,z\
...$ satisfying conditions 1)–4), then $G$ can be made into a totally ordered group with $P$ as set of positive elements.
3 KB (495 words) - 19:18, 17 August 2014
...e group generated by all right translations of a finite distributive quasi-group is solvable [[#References|[4]]].
...ign="top">[3]</TD> <TD valign="top"> V.D. Belousov, "Foundations of the theory of quasi-groups and loops" , Moscow (1967) (In Russian)</TD></TR>
3 KB (425 words) - 22:16, 7 January 2017
...oup algebra]]; [[Cross product|Cross product]]), with $|G|=q$. An $R$-Hopf order in $KG$ is a rank-$q$ $R$-Hopf algebra $H$ (cf. [[Hopf algebra|Hopf algebra
...(g)$. Then the $R$-Hopf order in $KG$ determined by $\xi$ (called a Larson order) is of the form
5 KB (919 words) - 04:20, 15 February 2024
...|nil semi-group]], or $S$ is a subdirect product of a group and a nil semi-group.
...nown (including necessary and sufficient ones) under which a periodic semi-group is a band of torsion classes; this clearly occurs for commutative semi-grou
5 KB (775 words) - 17:22, 17 October 2014
...lity in the lattice of all subgroups of a finite group (cf [[Supersolvable group]]).
<TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Hall, Jr., "The theory of groups" , Macmillan (1964) {{ZBL|0116.25403}}</TD></TR>
779 bytes (109 words) - 19:05, 1 May 2024
''Černikov group''
...also Chernikov; further, an extension of a Chernikov group by a Chernikov group is again Chernikov.
4 KB (589 words) - 18:19, 15 January 2017
''group algebra of a group $G$ over a field $K$''
...a; multiplication of basis elements in the group algebra is induced by the group multiplication. The algebra $KG$ is isomorphic to the algebra of functions
5 KB (872 words) - 22:00, 29 April 2012
A matrix group $G$ over an arbitrary fixed field $K$, all elements of which may be reduced
...mpletely reducible. Every normal subgroup of a completely-reducible matrix group is itself completely reducible.
2 KB (303 words) - 16:55, 10 April 2016
...e poly-nilpotent length is less than the solvable length. A poly-nilpotent group of length 2 is called meta-nilpotent.
...ncreasing) poly-nilpotent series of length $l$ whose factors in increasing order have nilpotent classes not exceeding the numbers $c_1,\dots,c_l$, respectiv
2 KB (229 words) - 16:12, 4 October 2014
The automorphism group of a
group of all automorphisms of the field $L$ leaving the elements of
3 KB (494 words) - 21:56, 5 March 2012
$#C+1 = 10 : ~/encyclopedia/old_files/data/S084/S.0804920 Shmidt group
...lpotent group]]). A Shmidt group is a [[Solvable group|solvable group]] of order $ p ^ \alpha q ^ \beta $,
2 KB (346 words) - 18:45, 11 April 2023
...G. Fubini in 1916 (a generalization of this concept to the geometry of any group of transformations was obtained by E. Cartan in 1920) using the concept of
be the group of transformations of a space $ E $.
3 KB (529 words) - 08:08, 6 June 2020
A group having a [[Normal series|normal series]]
...group itself), and their lengths are equal to the nilpotency class of the group.
3 KB (500 words) - 16:51, 30 December 2018