Are All Indecomposable Groups Cyclic?

[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!

 

[Stephen Tashi]

Look at a list of "simple" groups.

 

[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/%7Eroy/4050sum08.pdf

 

[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.

 

[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.