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1. Subgroup: A non-empty subset $H$ of a group $G$ which itself is a group with respect to the operation defined on $G$. A subset $H$ of a group $G$ is a subgroup of $G$ if and only if: 1) $H ... (Mathematics) [100%] 2024-01-26
2. Subgroup: In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a ... (Subset of a group that forms a group itself) [100%] 2024-01-26 [Group theory] [Subgroup properties]...
3. Subgroup: In group theory, a subgroup of a group is a subset which is itself a group with respect to the same operations. Formally, a subset S of a group G is a subgroup if it satisfies the following conditions: These ... [100%] 2023-10-25
4. Subgroup: A non-empty subset $H$ of a group $G$ which itself is a group with respect to the operation defined on $G$. A subset $H$ of a group $G$ is a subgroup of $G$ if and only if: 1) $H ... (Mathematics) [100%] 2023-10-21
5. Subgroup: In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a ... (Subset of a group that forms a group itself) [100%] 2023-12-30 [Group theory] [Subgroup properties]...
6. Verbal subgroup: The subgroup $ V( G) $ of a group $ G $ generated by all possible values of all words (cf. Word) of some set $ V = \{ {f _ \nu ( x _ {1} \dots x _ {n _ \nu } ) } : {\nu \in I } \} $, when $ x _ ... (Mathematics) [70%] 2023-10-17
7. Characteristic subgroup: A subgroup of a group $G$ that is invariant under all automorphisms of $G$. Examples of characteristic subgroups are the centre of a group, denoted by $Z(G)$, the Fitting subgroup, $F(G)$, the commutator subgroup, $D(G)$, $[G,G ... (Mathematics) [70%] 2023-10-17
8. Parabolic subgroup: A parabolic subgroup of a linear algebraic group $G$ defined over a field $k$ is a subgroup $P\subset G$, closed in the Zariski topology, for which the quotient space $G/P$ is a projective algebraic variety. A subgroup $P ... (Mathematics) [70%] 2023-12-30
9. Subgroup system: A set $ \mathfrak A $ of subgroups (cf. Subgroup) of a group $ G $ satisfying the following conditions: 1) $ \mathfrak A $ contains the unit subgroup $ 1 $ and the group $ G $ itself; and 2) $ \mathfrak A $ is totally ordered by inclusion, i.e. (Mathematics) [70%] 2023-10-24
10. Commutator subgroup: The commutator subgroup of a group $G$, also called the derived group, second term of the lower central series, of $G$, is the subgroup of $G$ generated by all commutators of the elements of $G$ (cf. Commutator). (Mathematics) [70%] 2023-08-16
11. Congruence subgroup: In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the off ... (Matrix group) [70%] 2023-04-16 [Modular forms] [Discrete groups]...
12. Puig subgroup: In mathematical finite group theory, the Puig subgroup, introduced by Puig (1976), is a characteristic subgroup of a p-group analogous to the Thompson subgroup. If H is a subgroup of a group G, then LG(H) is the subgroup ... [70%] 2023-12-02 [Finite groups]
13. Subgroup series: A subgroup series of a group $G$ is a finite chain of subgroups of $G$ contained in each other: $$E=G_0\subseteq G_1\subseteq \cdots \subseteq G_n=G\label{1}$$ or $$G=G_1\supseteq G_2\supseteq \cdots \supseteq G_{n ... (Mathematics) [70%] 2023-12-29
14. Frattini subgroup: In group theory, the Frattini subgroup is the intersection of all maximal subgroups of a group. Formally, where M runs over all maximal subgroups of G. [70%] 2023-09-10
15. Verbal subgroup: The subgroup $ V( G) $ of a group $ G $ generated by all possible values of all words (cf. Word) of some set $ V = \{ {f _ \nu ( x _ {1} \dots x _ {n _ \nu } ) } : {\nu \in I } \} $, when $ x _ ... (Mathematics) [70%] 2024-01-26
16. Characteristic subgroup: A subgroup of a group $G$ that is invariant under all automorphisms of $G$. Examples of characteristic subgroups are the centre of a group, denoted by $Z(G)$, the Fitting subgroup, $F(G)$, the commutator subgroup, $D(G)$, $[G,G ... (Mathematics) [70%] 2024-01-26
17. Parabolic subgroup: A parabolic subgroup of a linear algebraic group $G$ defined over a field $k$ is a subgroup $P\subset G$, closed in the Zariski topology, for which the quotient space $G/P$ is a projective algebraic variety. A subgroup $P ... (Mathematics) [70%] 2024-01-26
18. Subgroup series: A subgroup series of a group $G$ is a finite chain of subgroups of $G$ contained in each other: $$E=G_0\subseteq G_1\subseteq \cdots \subseteq G_n=G\label{1}$$ or $$G=G_1\supseteq G_2\supseteq \cdots \supseteq G_{n ... (Mathematics) [70%] 2024-01-26
19. Subgroup method: The subgroup method is an algorithm used in the mathematical field of group theory. It is used to find the word of an element. [70%] 2024-01-26 [Group theory]
20. Pure subgroup: In mathematics, especially in the area of algebra studying the theory of abelian groups, a pure subgroup is a generalization of direct summand. It has found many uses in abelian group theory and related areas. [70%] 2024-01-26 [Subgroup properties] [Abelian group theory]...