Algebra by Larry C. Grove

By Larry C. Grove

This graduate-level textual content is meant for preliminary classes in algebra that continue at a speedier velocity than undergraduate-level classes. workouts seem during the textual content, clarifying recommendations as they come up; extra routines, various generally in hassle, are incorporated on the ends of the chapters. topics contain teams, earrings, fields and Galois thought. 1983 version. contains eleven figures. Appendix. References. Index.

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If G is not abelian show that Z(G)is properly contained in an abelian subgroup of G. 5. If G is a group and 1x1 = 2 for all x # 1 in G show that G is abelian. Can you say more? 6. Suppose G is a group, H I G, and K I G. Show that H v K is not a group unless H I K or K I H. 7. (Haber and Rosenfeld [12]). Show that a group G is the union of three proper subgroups if and only if there is an epimorphism from G to Klein’s 4-group. 8. Suppose S is a subset of a finite group G, with IS[ > ICl/2. If S 2 is defined to be { x y : x ,y E S } show that S z = G.

2,- 2 Corollary. Finite p-groups are solvable. 1. Show by example that a solvable group need not be nilpo tent. 3. If G is nilpotent and H I G but H # G,then N,(H) # H . Proof. Choose Z i in the ascending central series such that Zi5 H but 4 H . We show that Z i + ' 5 NG(H). If x E Zi+l, then x Z i E Z i + , / Z i= Z(G/Zi), so for any h E H we have x Z i h - 'Zi= h - ' Z i x Z i , and hence x 'hxh ' E Z i S H . Consequently x - hx E H , and so x E NG(H). ~ ' Corollary. If G is nilpotent and H I G is a maximal proper subgroup, then H 4 G .

If a universal pair ( U , E ) exists for a group G then U is unique (up to isomorphism). 5 Solvable Groups, Normal and Subnormal Series Proof. or E~ = glE Let ( U , ,E and ~ be ) 23 another universal pair for G. We have E = g2El. Thus E~ = g1g2s, and E = g2glE. But then we have and by the uniqueness in the definition of a universal pair we see that g2gl = l,,, the identity map on U . Similarly glg2 is the identity map on U , , and so g1 and g 2 are inverse isomorphisms. 1. If ( U , E ) is a universal pair for a group G and h E Aut(U) show that ( U , he) is also universal for G.

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