Does Preimage of Subgroup Under Homomorphism Form a Subgroup?

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Suppose θ: A → B is a homomorphism. And assume S ≤ B. Is it necesarily true that if S is a subgroup, that is not completely contained in the range, its preimage forms a subgroup?
 
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I used subgroup criterion test, and it should be a subgroup. But i just wanted to make sure I didn't miss anything trivial.
 
xiavatar said:
I used subgroup criterion test, and it should be a subgroup. But i just wanted to make sure I didn't miss anything trivial.

You didn't. It's a subgroup indeed!
 

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