Are All Indecomposable Groups Cyclic?

1. Sep 23, 2014

metapuff

A group is said to be indecomposable if it cannot be written as a product of smaller groups. An example of this is any group of prime order p, which is isomorphic to the group of integers modulo p (with addition as the group operation). Since the integers modulo p is a cyclic group (generated by 1), we have that any indecomposable group of prime order is cyclic. I have two questions:

Are ALL indecomposable groups cyclic? (N.B. not just those of prime order)

Are all cyclic groups indecomposable?

Thanks!

2. Sep 24, 2014

Stephen Tashi

Look at a list of "simple" groups.

3. Sep 28, 2014

mathwonk

If you restrict to finite abelian groups, then all indecomposable such groups are cyclic, but not all cyclic groups are indecomposable. All finite abelian groups decompose into a product of not just cyclic factors, but of indecomposable cyclic factors.

see my notes, especially pages 42-43:

http://www.math.uga.edu/~roy/4050sum08.pdf

4. Sep 28, 2014

metapuff

Ah, thanks @mathwonk! Your notes turned out to be just the reference I was looking for. I thought the extension to linear mappings was pretty interesting.

5. Oct 2, 2014

mathwonk

you are welcome. i had fun thinking that stuff through for a summer course in linear algebra, especially the analogies with finite abelian groups.

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