A Course on Group Theory
This textbook for advanced courses in group theory focuses on finite groups, with emphasis on the idea of group actions. Early chapters summarize presupposed facts, identify important themes, and establish the notation used throughout the book. Subsequent chapters explore the normal and arithmetical structures of groups as well as applications.
Topics include the normal structure of groups: subgroups; homomorphisms and quotients; series; direct products and the structure of finitely generated Abelian groups; and group action on groups. Additional subjects range from the arithmetical structure of groups to classical notions of transfer and splitting by means of group action arguments. More than 675 exercises, many accompanied by hints, illustrate and extend the material.
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action of G argue by induction Aut G automorphism CG(x characteristic subgroup composition series conjugacy class cyclic group deﬁned deﬁnition denote direct product distinct primes element of G factor of G ﬁeld ﬁnite group ﬁrst follows G acts G is ﬁnite G is nilpotent G is soluble G of order group G group of order H in G Hall subgroup Hence Hint homomorphism identity element involution isomorphic K Q G Lemma Let G Let H Q G Let K Q Math maximal subgroup minimal normal subgroup nilpotent groups normal in G p-group permutation positive integer prime divisor Proof Let proper subgroup prove quotient group series of G soluble group subgroup H subgroup of G subgroup of index subgroup of order subnormal in G subnormal subgroup Suppose that G Sylow p-subgroup Sylow’s theorem transversal to H trivial vector space wreath product