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 = ∪{ xHx1  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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 Fall '08
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 Algebra, Normal subgroup, Abelian group, Subgroup, Cyclic group, Dr. Matthew Macauley

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