Math 720

Fall 2000

Homework 3

1. Show that any finite group, G, can be embedded in S_n (that is, is a subgroup of S_n) for some n.

2. Compute:

a) All normal subgroups of D_n.

b) A subgroup of order 20 in S₅.

c) Is there a subgroup of order 40 in S₅?

3. Show that S_n is generated by two elements.

4. Show that S₃ is not the direct product of any family of its proper subgroups.

5. Let G be abelian with subgroups H and K. Show that if and only if there exist homomorphisms:

Such that

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6. It is true that if H and G are groups and and Then Is it true that if H and G are groups with and G isomorphic to a subgroup of H is it true that