Let G be a group and be a subgroup of G such that H notequal

Let G be a group and be a subgroup of G such that H notequalto G. It is said that His a maximal subgroup of G if there is no subgroup F of G such that H F and F notequalto H, G. Similarly, H is said to be a maximal normal subgroup of G if H G and there is no normal subgroup F of G such that H F and F notequalto H, G. Prove that Q has no maximal subgroups.

Solution

H is a maximal normal subgroup of G if and only if G/H is simple.

Let, A/HG/H

wherein HAG.

Since H is a maximal subgroup, H=A or A=Gand so, A/H=1 or A/H=G/H. This means that G/H is a simple group. Now suppose that HG and G/H is simple.

If we have HAG. then obviously A/HG/H and that G/H is simple, we get A/H=G/H or A/H={H}. So, A=G or H=A.