Question:Normal subgroup quotient group determine whole group

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Q: Does knowing the isomorphism type of a normal subgroup and of the quotient group determine the isomorphism type of the whole group?

A: The short answer is not in general.

Given the isomorphism class of the normal subgroup and the isomorphism class of the quotient group, there is always at least one candidate for the original group -- an external direct product of these two groups, where we then identify the normal subgroup with the first direct factor. Specifically, in the external direct product of groups A \times B, the subgroup A \times \{ e \} is isomorphic to A and the quotient group by this is isomorphic to B. (see also internal direct product and equivalence of internal and external direct product).

However, it is possible to have groups with a normal subgroup isomorphic to A and a quotient group isomorphic to B, but where the group is not isomorphic to A \times B. The smallest example is where the whole group is cyclic of order four. This has a subgroup of order two (comprising the identity element and the square of the generator) that is isomorphic to the cyclic group of order two, and the quotient is also isomorphic to the cyclic group of order two.

However, the cyclic group of order four is not isomorphic to the direct product of two copies of the cyclic group of order two. The direct product is the Klein four-group, which is a different isomorphism class.

An easy way to see this is that in the cyclic group of order four, there is no element of order two outside the subgroup of order two, so the subgroup of order two does not have a complement of order two. We need a complement in order to obtain an internal direct product decomposition.

Advanced: The problem of finding all possible groups having a normal subgroup of specified isomorphism class and a quotient group of specified isomorphism class is called the extension problem. An important tool in this problem is the second cohomology group, which is a group that stores the possible extensions in some special cases and given some additional data about the nature of the extension.

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