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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.
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 group 2 F 4 (2) also occurs as a maximal subgroup of the Rudvalis group, as the point stabilizer of the rank-3 permutation action on 4060 = 1 + 1755 + 2304 points. The Tits group is one of the simple N-groups , and was overlooked in John G. Thompson 's first announcement of the classification of simple N -groups, as it had not been ...
Since the normal subgroup is a subgroup of H, its index in G must be n times its index inside H. Its index in G must also correspond to a subgroup of the symmetric group S n, the group of permutations of n objects. So for example if n is 5, the index cannot be 15 even though this divides 5!, because there is no subgroup of order 15 in S 5.
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).
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 ...
A group action is primitive if it is isomorphic to G/H for a maximal subgroup H of G, and imprimitive otherwise (that is, if there is a proper subgroup K of G of which H is a proper subgroup). These imprimitive actions are examples of induced representations.