Let H be a subgroup of index 2 of a group G Prove that H mus

Let H be a subgroup of index 2 of a group G. Prove that H must be a normal subgroup of G. Conclude that S_n is not simple for n 3.

Solution

Let G be a group and H be a subgroup of index 2.

Then H partitions G into 2 left cosets H and aH , and similarly H partitions G into 2 right cosets, H ,Ha.

If a H then aH = H = Ha since H is a subgroup of G.

If a G - H then aH = G - H = Ha.

Thus, aH = Ha for all a G and H is normal sub group of G.

A finite group is called simple when its only normal subgroups are the trivial subgroup and the whole group.

When n 3 the group S_n is not simple because it has a nontrivial normal subgroup A_n. But the groups A_n are simple, provided n >5.

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