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).
When n is a positive integer, Sz(q) has at least 4 types of maximal subgroups. The diagonal subgroup is cyclic, of order q – 1. The lower triangular (Borel) subgroup and its conjugates, of order q 2 ·(q-1). They are one-point stabilizers in a doubly transitive permutation representation of Sz(q).
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.
The full automorphism group of A 6 appears naturally as a maximal subgroup of the Mathieu group M 12 in 2 ways, as either a subgroup fixing a division of the 12 points into a pair of 6-element sets, or as a subgroup fixing a subset of 2 points.
All maximal tori in G are conjugate. [3] All maximal tori have the same dimension, known as the rank of G. 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)
It does not preserve the Lie bracket of this lattice, but does preserve the Lie bracket mod 3, so is a subgroup of the Chevalley group E 8 (3). The subgroup preserving the Lie bracket (over the integers) is a maximal subgroup of the Thompson group called the Dempwolff group (which unlike the Thompson group is a subgroup of the compact Lie group ...
G 0 is the trivial subgroup; G i is a normal subgroup of G i+1 (for every i between 0 and n - 1) and the quotient group G i+1 / G i is a cyclic group (for every i between 0 and n - 1) A metacyclic group is a polycyclic group with n ≤ 2, or in other words an extension of a cyclic group by a cyclic group.