# Search results

• The Group Properties Wiki is designed and organized using a property-theoretic paradigm. This ar Examples from group theory:
7 KB (1,067 words) - 22:24, 7 May 2008
• ...ties over structures and the algebraic and logical relations between those properties'' Examples from group theory:
4 KB (575 words) - 22:44, 7 May 2008
• This is an informal account of the story of the development of the Group Properties Wiki, till 2007. The '''Group Properties Wiki''', called ''groupprops'' first began life at
2 KB (398 words) - 22:44, 7 May 2008
• ...lves whatever problem we are considering in terms of oracle queries to the group operations and any pre-processing or post-processing that needs to be done. Any black-box group algorithm gives an actual algorithm if we are actually provided an encoding
2 KB (408 words) - 07:21, 21 December 2014
• ...nsion $F$ over $\mathbb{Q}$ satisfying the following properties. Let $I$ denote the ring of algebraic integers in $F$
2 KB (401 words) - 23:12, 7 May 2008
• An element of a [[group]] is termed central if the following equivalent conditions hold: # It commutes with every element of the group
15 KB (2,081 words) - 20:14, 1 June 2016
• Let $\sigma$ be an automorphism of a group $G$. We say that $\sigma$ is '''characteristic-semidi ===Stronger properties===
1 KB (150 words) - 15:44, 8 April 2009
• A [[subgroup]] of a [[group]] is termed a '''characteristic direct factor''' if it is a [[characteristi * Every group is a characteristic direct factor of itself.
4 KB (455 words) - 02:59, 20 January 2013
• ...core]] is a [[defining ingredient::characteristic subgroup]] of the whole group. A [[subgroup]] $H$ of a [[group]] $G$ is termed '''core-characteristic''' if the normal core <ma
3 KB (285 words) - 17:34, 21 December 2014
• A prime divisor of a number is said to be '''core-nontrivial''' if for every group whose order is that number, the core for that prime divisor is nontrivial. ...nontrivial''' if for any group $G$ of order $N$, the group $O_p(G)$ (viz the $p$-core of $G$) is nont
2 KB (413 words) - 23:24, 7 May 2008
• {{prime-parametrized sdf}} Let $G$ be a [[finite group]] of order $N$ and let $p$ be a prime divisor of <mat
2 KB (287 words) - 23:24, 7 May 2008
• A family $\mathcal{F}$ of subgroups of a group $G$ is termed a '''cover''' for $G$ if it satisfies t ==Properties==
876 bytes (150 words) - 23:24, 7 May 2008
• {{nottobeconfusedwith|[[critical group]]}} ...bgroup''' if it is [[characteristic subgroup|characteristic]] in the whole group and satisfies the following three conditions:
4 KB (489 words) - 00:05, 13 February 2009
• {{pivotal group property}} {{natural number-parametrized group family}}
8 KB (1,111 words) - 23:25, 20 June 2013
• {{particular group}} ...ined as the unique group of order two. Explicitly it can be described as a group with two elements, say $e$ and $x$ such that $ex 8 KB (1,234 words) - 05:31, 21 January 2013 • {{particular group}} ...ined as the unique group of order 3. Equivalently it can be described as a group with three elements [itex]e= x^0,x,x^2$ where $x^lx^m = x^{l+m}< 6 KB (839 words) - 04:07, 12 January 2013 • {{prime set-parametrized group property}} Let [itex]G$ be a [[finite group]], and $\pi$ be a set of primes. We say that $G$ sati
821 bytes (134 words) - 23:25, 7 May 2008
• {{nottobeconfusedwith|[[dicyclic group]] (also called binary dihedral group)}} {{group property}}
10 KB (1,448 words) - 02:22, 6 July 2019
• {{particular group}} ...times the ''octic group'', is defined by the following [[presentation of a group|presentation]], with $e$ denoting the identity element:
19 KB (2,660 words) - 13:15, 14 February 2015
• ...up correlates with the way the normal subgroups are distributed within the group? This, and other related questions, are explored within this article. ! Group !! Order !! Orders of normal subgroups (with multiplicity) !! Indexes of no
11 KB (1,600 words) - 23:24, 3 November 2010

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