Search results
Results from the WOW.Com Content Network
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 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).
An octern is a certain partition of the 24 points into 8 blocks of 3. The subgroup fixing an octern is the octern group isomorphic to PSL(2,7), of order 168, simple, transitive and imprimitive. It was the last maximal subgroup of M 24 to be found. The table below lists all the maximal subgroups.
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.
In the second row are the maximal subgroups; their intersection (the Frattini subgroup) is the central element in the third row. So Dih 4 has only one non-generating element beyond e . In mathematics , particularly in group theory , the Frattini subgroup Φ ( G ) {\displaystyle \Phi (G)} of a group G is the intersection of all maximal subgroups ...
A maximal torus in G is a maximal abelian subgroup, but the converse need not hold. [4] The maximal tori in G are exactly the Lie subgroups corresponding to the maximal abelian subalgebras of [5] (cf. Cartan subalgebra) Every element of G lies in some maximal torus; thus, the exponential map for G is surjective.
The unitary group is a subgroup of the general linear group GL(n, C), and it has as a subgroup the special unitary group, consisting of those unitary matrices with determinant 1. In the simple case n = 1, the group U(1) corresponds to the circle group, isomorphic to the set of all complex numbers that have absolute value 1, under multiplication ...
The automorphism group of the complex Leech lattice is the universal cover 6 · Suz of the Suzuki group. This makes the group 6 · Suz · 2 into a maximal subgroup of Conway's group Co 0 = 2 · Co 1 of automorphisms of the Leech lattice