homework6 - a,b A and b = g.a for some g G , then G b = gG...

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Math 113 Section 5 Homework #6 Fall 2010 Due: Monday, October 18 1. Prove the Third Isomorphism Theorem: Let G be a group and let H and K be normal subgroups of G such that H is a subgroup of K . Then K/H is a normal subgroup of G/H and ( G/H ) / ( K/H ) is isomorphic to G/K . 2. Let G be a group and let N be a normal subgroup. Let π N : G G/N be the canonical projection and let H < G/N . Show that π - 1 ( H ) is a subgroup of G . 3. Prove the first statement of the Fourth Isomorphism Theorem (Correspondence Theo- rem): Let G be a group, N a normal subgroup of G . Show that the map θ : { H < G : N < H } → { H/N < G/N } , θ ( H ) = H/N is a bijection. 4. Section 3.3, Exercise 7: Let M and N be normal subgroups of G such that G = MN . Prove that G/ ( M N ) is isomorphic to ( G/M ) × ( G/N ) . (Hint: draw the lattice.) 5. Section 3.5, Exercise 12: Prove that A n contains a subgroup isomorphic to S n - 2 for each n 3 . 6. Section 4.1, Exercises 1, 2 and 3: (a) Let G act on the set A . Prove that if
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Unformatted text preview: a,b A and b = g.a for some g G , then G b = gG a g-1 ( G a is the stabilizer of a ). Deduce that if G acts transitively on A then the kernel of the action is g G gG a g-1 . (b) Let G be a permutation group on the set A (i.e., G &lt; S A ), let G and let a A . Prove that G a -1 = G ( a ) . Deduce that if G acts transitively on A then G G a -1 = { e } . (c) Assume that G is an abelian, transitive subgroup of S A . Show that ( a ) = a for all G- { e } and for all a A . Deduce that | G | = | A | . (Hint: use the preceding exercise.) 7. Section 4.2, Exercise 4: Use the left regular representation of the quaternionic group Q 8 to produce two elements of S 8 which generate a subgroup of S 8 isomorphic to Q 8 . 1...
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