# Every subgroup of order 74 in a group of order 148 is normal

Every subgroup of order 74 in a group of order 148 is normal

Discussion:

True

We will prove a much general claim: if index of a subgroup is 2, then that subgroup must be normal.

Suppose $$H \le G$$ and $$[G:H] = 2$$ .

Now, if $$g \in H$$ then gH = Hg = H

Otherwise if $$g \not\in H$$ then gH = G H  (why? because we know that any two cosets are completely distinct or entirely equal, and number of elements in gH is same as number of elements in H; prove this)

But Hg = GH as well.

So gH = Hg when g is not in H as well.

Hence H is normal subgroup of G.

Back to problem list