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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-  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.]
-  Let g be an element of a group G.
- (a) If x Î
<g>, show that
<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>.
-  Let S, S1 and S2 be subsets of a group G.
- (a) If S1 Ì S2, show that
CG(S2) Ì CG(S1).
- (b) Show that CG(S) =
Çs ÎS CG(s).
-  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|.
-  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.