Here we construct a group of order 36 which has a nonnormal subgroup of order 12. Let S₃ denote the symmetric group of degree 3. Then is a group of order 36 which has a normal subgroup K such that (e.g., we could let ). Now S₃, and hence also G/K, have a nonnormal subgroup of order 2. Using the subgroup correspondence theorem, we deduce that G has a nonnormal subgroup of order (which contains K), as required.

• Spring 1981, prob. 3

 
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