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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.

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April 14, 2015

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