Maximal Subgroups of Finite Groups
نویسنده
چکیده
What ingredients are necessary to describe all maximal subgroups of the general finite group G? This paper is concerned with providing such an analysis. A good first reduction is to take into account the first isomorphism theorem, which tells us that the maximal subgroups containing a given normal subgroup N of G correspond, under the natural projection, to the maximal subgroups of the quotient group GIN. Let n = nG denote the collection of maximal subgroups of G, and let n * be the subset of those MEn with KerG(M) = 1, where KerG(M) denotes the largest normal subgroup of G contained in M. Then the first isomorphism theorem allows us to identify n with the disjoint union UN <lG nS/N' Actually, what we really want to parameterize are the conjugacy classes of maximal subgroups, but this too works well: If W = WG denotes the set of G-conjugacy classes of elements of n, and W* is defined similarly, then we have
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