Group Theory: Math30038, Sheet 8

GCS

1. Let P be a Sylow p-subgroup of the finite group G. Suppose that N ⊴G. (a) Show that P ∩ N ∈ Syl_{p}(N).

(b) Show that P N/N ∈ Syl_{p}(G/N).

2. Show that every group of order 15 must be abelian.

3. Show that every group of order 35 must be abelian.

4. Show that there is no non-abelian finite simple group of order less than 60.

5. Let p and q be distinct prime numbers.

(a) Show that a group of order pq can not be simple.

(b) Show that a group of order p^{2}q can not be simple.

(c) Show that a group of order p^{2}q^{2} can not be simple.