Math 417 Homework 4: Due Friday February 14

Instructions: Do any four of the five problems. You can discuss these problems with others, but write up your solutions on your own (i.e., don't just copy someone else's solutions, else the feedback I give you won't help you much). Please be neat and write in full sentences.

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[1] Let H and K be subgroups of a group G. Prove that HÇK is a subgroup of G. [Note: the symbol Ç should be an intersection symbol.]
[2] Let g be an element of a group G.
- (a) If x Î <g>, show that <x> Ì <g> and hence that |x| £ |g|. [Note: the symbol Î should be "is an element of" symbol; the symbols < and > should be the symbols showing that <g> is the cyclic group generated by g; the symbol £ should be a "less than or equals" symbol; and the symbol Ì should be "is a subset of" symbol.]
- (b) Use (a) to conclude that <g^-1> = <g>.
[3] Let S, S₁ and S₂ be subsets of a group G.
- (a) If S₁ Ì S₂, show that C_G(S₂) Ì C_G(S₁).
- (b) Show that C_G(S) = Ç_{s ÎS} C_G(s).
[4] Let G be a group which has exactly three different subgroups, including a proper subgroup H of order 7. Show that G is cyclic, and determine |G|.
[5] Let G be a cyclic group of order n. Let D and M be subgroups of G of orders d and m respectively. Determine the order of D Ç M, in terms of d, m and n.