Thread: Find a non subgroup out of subgroups cosets

Find a group G with subgroups A and B such that AB is not a subgroup.

I think I may be overthinking this, let G be the set of integers, then let A be {1} and B = {0}, then both A and B are subgroup, but there would be no inverse for 0 in the set AB.

Is this right?

Thank you.

Originally Posted by tttcomrader

Find a group G with subgroups A and B such that AB is not a subgroup.

I think I may be overthinking this, let G be the set of integers, then let A be {1} and B = {0}, then both A and B are subgroup, but there would be no inverse for 0 in the set AB.

Is this right? no! the set of integers is an additive group and so {1} is not a subgroup.

you know that if one of A and B is normal, then AB will be a subgroup. so the group G that you're looking for has to be non-abelian. there are many examples. here are two of them:

Example 1 G finite: let then which doesn't divide so AB is not a subgroup.

Example 2 G infinite: let let then now

but so AB is not a subgroup.

If you are looking for an infinite case counterexample it is easier just to consider .
Now choose and .