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



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.

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