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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>.

- (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.]
- [3] Let S, S
_{1}and S_{2}be subsets of a group G.- (a) If S
_{1}Ì S_{2}, show that C_{G}(S_{2}) Ì C_{G}(S_{1}).

- (b) Show that C
_{G}(S) = Ç_{s ÎS}C_{G}(s).

- (a) If 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.