Thread: TA’s Fun Algebra Problem

Hi all.

Following NonCommAlg’s example, I have decided to post a fun algebra (group theory) problem of my own on this forum.

Let where is an abelian group and is a cyclic group of order 2, and suppose contains an element with Find a binary operation on such that is a nonabelian group.

To NonCommAlg: If you know the answer to this one (and I have a feeling you do) please don’t be too quick to reveal the answer. Let others have a chance to do the problem first.

Originally Posted by TheAbstractionist

Hi all.

Following NonCommAlg’s example, I have decided to post a fun algebra (group theory) problem of my own on this forum.

Let where is an abelian group and is a cyclic group of order 2, and suppose contains an element with Find a binary operation on such that is a nonabelian group.

To NonCommAlg: If you know the answer to this one (and I have a feeling you do) please don’t be too quick to reveal the answer. Let others have a chance to do the problem first.

it took me some time to figure this one out!

Spoiler:

let define by and now define by and

then is a group because and it's non-abelian because for any with we have:

Originally Posted by NonCommAlg

it took me some time to figure this one out!

Spoiler:

let define by and now define by and

then is a group because and it's non-abelian because for any with we have:

Hi NonCommAlg.

Yes, that was what I had in mind.

For those who don’t understand NonCommAlg’s proof:

Spoiler:

The binary operation is this: For

Comments:

Spoiler:

You can verify that all the group axioms are satisfied for the operation defined above, but checking associativity directly can be rather time consuming. Fortunately, we have a theorem.

Let be any group, a cyclic group of order and an automorphism of such that is the identity automorphism. Then is a group with respect to the operation where

for all and

is known as the cyclic extension of by induced by the automorphism For a proof of the theorem (and more on cyclic extensions) see Chapter 21 of John F. Humphreys, A Course in Group Theory, Oxford University Press, 1996.

In our example, is the mapping which is an automorphism because is Abelian. And is not Abelian because whereas the two are not equal if