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In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra. In group theory, a maximal subgroup H of a group G is a proper subgroup, such that no proper subgroup K contains H strictly. In other words, H is a maximal element of the partially ordered set of subgroups of G that are not equal to G.
This follows from inspection of 5-cycles: each 5-cycle generates a group of order 5 (thus a Sylow subgroup), there are 5!/5 = 120/5 = 24 5-cycles, yielding 6 subgroups (as each subgroup also includes the identity), and S n acts transitively by conjugation on the set of cycles of a given class, hence transitively by conjugation on these subgroups.
A maximal compact subgroup is a maximal subgroup amongst compact subgroups – a maximal (compact subgroup) – rather than being (alternate possible reading) a maximal subgroup that happens to be compact; which would probably be called a compact (maximal subgroup), but in any case is not the intended meaning (and in fact maximal proper subgroups are not in general compact).
A large subgroup H (preferably a maximal subgroup) of the Monster is selected in which it is easy to perform calculations. The subgroup H chosen is 3 1+12.2.Suz.2, where Suz is the Suzuki group. Elements of the monster are stored as words in the elements of H and an extra generator T.
A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is often represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e} ). [2] [3] If H is a subgroup of G, then G is sometimes called an overgroup of H.
The intransitive maximal subgroups are exactly those of the form S k × S n–k for 1 ≤ k < n/2. The imprimitive maximal subgroups are exactly those of the form S k wr S n/k, where 2 ≤ k ≤ n/2 is a proper divisor of n and "wr" denotes the wreath product.
A maximal torus in a compact Lie group G is a maximal subgroup among those that are isomorphic to T k for some k, where T = SO(2) is the standard one-dimensional torus. [2] In O(2n) and SO(2n), for every maximal torus, there is a basis on which the torus consists of the block-diagonal matrices of the form
Let G 0 be the connected component of the identity, i.e., the maximal connected subgroup scheme. Then G is an extension of a finite étale group scheme by G 0. G has a unique maximal reduced subscheme G red, and if k is perfect, then G red is a smooth group variety that is a subgroup scheme of G. The quotient scheme is the spectrum of a local ...