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 and .

Now, if then gH = Hg = H

Otherwise if 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)