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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 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).
For a prime number, a Sylow p-subgroup (sometimes p-Sylow subgroup) of a finite group is a maximal-subgroup of , i.e., a subgroup of that is a p-group (meaning its cardinality is a power of ; or equivalently: For each group element, its order is some power of ) that is not a proper subgroup of any other -subgroup of .
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.
In a directed set, every pair of elements (particularly pairs of incomparable elements) has a common upper bound within the set. If a directed set has a maximal element, it is also its greatest element, [proof 7] and hence its only maximal element. For a directed set without maximal or greatest elements, see examples 1 and 2 above.
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. The numbers of primitive groups of small degree were stated by Robert Carmichael in 1937:
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.
In mathematics, particularly in group theory, the Frattini subgroup of a group G is the intersection of all maximal subgroups of G. For the case that G has no maximal subgroups, for example the trivial group { e } or a Prüfer group , it is defined by Φ ( G ) = G {\displaystyle \Phi (G)=G} .