s09_mthsc851_hw01 - | xy | = | yx |(7(a Prove that if G/Z G...

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MTHSC 851 (Abstract Algebra) Dr. Matthew Macauley HW 1 Due Tuesday Jan. 20, 2009 (1) Let G be a finite group and n > 2. Show that the number of elements in G of order n is even. (2) Show that every group of even order has an element of order 2. (3) If a group G has a unique element x of order 2 show that x Z ( G ). (a) Show that any finitely-generated subgroup of ( Q , +) (the additive group of the ratio- nals) is cyclic. Use this to show that ( Q , +) is not isomorphic to ( Q , +) ( Q , +) (the direct sum of two copies of Q .) (b) What happens if Q is replaced by R in both parts of (a)? (4) Suppose G is a group, H G , and K G . Give necessary and sufficient conditions for H K to be a group. (5) 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. (6) (a) Suppose H G . Show that gHg - 1 is a subgroup, and that H = gHg - 1 . (b) Use (a) to show that in any group,
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Unformatted text preview: | xy | = | yx | . (7) (a) Prove that if G/Z ( G ) is cyclic, then G is abelian (b) Prove that if Z ( G ) is maximal among abelian subgroups, then G is abelian. (8) Let | G | = ∞ , and [ G : H ] < ∞ . Show that H intersects every infinite subgroup of G nontrivially. (9) Suppose G is finite, H ≤ G , and G = ∪{ xHx-1 | x ∈ G } . Show that H = G . (10) Prove or give a counter example to each statement: (a) If every proper subgroup H of a group G is cyclic, then G is cyclic. (b) If H is a subgroup of an abelian group G , then both H and the quotient group G/H are abelian. (c) If H is a normal abelian subgroup of a group G , and the quotient group G/H is also abelian, then G is abelian. (d) If K C H C G , then K C G ....
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