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Abstract Algebra: Direct Products
Posted on 2008-10-29
I have two problems that i need help with! I'm trying to study for an exam and cannot figure out these review problems
first problem:
The group S3 (+) Z2 is isomorphic to one of the following groups: Z12,
Z6 (+) Z2, A4, D6. Determine which one by elimination.
I know to be isomorphic denoted phi from a group G to a group G' is a one to one mapping from G onto G' that preserves the group operation. So phi(ab)= phi(a) phi(b). I also know to prove one is isomorphic, need to have mapping (define a function phi from G to G'), need to be 1 to 1 (assume that phi(a)=phi(b) and prove that a=b), needs to be onto (prove for any element g' in G', find an element g in G such that phi(g)=g'), and prove that phi is operation preserving (show phi(ab)= phi(a)phi(b)).
I also know that S3 (+) Z2 the first group S3 is multiplicative and second group Z2 is additive so dealing with mixed binary operations.
I think S3 (+) Z2 = {(1), 0), ((1 2), 0), ((1 2 3), 1)}
then ((1 2, 0) * ((1 2 3), 1)= ((1 2)(1 2 3), 1)= ((1)(2 3 1), 1). But i'm not too sure.
second problem:
Find a subgroup of Z12 (+) Z4 (+) Z15 that is of order 9.
I know that the direct product is the set of all n-tuples for which the ith componenet is an element of Gi and the operation is componentwise.
I'm confused how to find direct products, can someone help me? Thanks so much, greatly appreciated!