If H and K are subgroups of G, Show tht H intersecting with K is a subgroup of G.  (Can you see that the same proof shows that the intersection of any number of subgroups of G, finitie or infinite, is again a subgroup of G?)

1. Show that, the order of any subgroup of a finite group divides the order of the group. (This proof has to be given after first showing that the cosets are either identical or disjoint)

4) Let H and K be two subgroups of a group, G. We have seen that HUK is not, in general, a subgroup of G. Show that if G-Z then the union of any two subgroups of G would, indeed, be a subgroup of G. [Hint: List all of the subgroups of 21

5. Prove that the intersection of two subgroups of G is itself a subgroup of G.

Let H be a subgroup of Sn. (a) Show that either all the permutations in H are even, or else half the permutations in H are even and half are odd. (b) Show that the set of even permutations in H form a subgroup of H.

Prove that a normal subgroup must be a union of conjugacy classes.

Please do the following questions with full handwritten working out.

17. With H and K as in Exercise 16, prove that HK is a subgroup of G.