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finite abelian group

Contents

Definition

A finite abelian group is a group which is both finite and abelian.

Properties

Proposition

If a finite abelian group AA has order |A|=p{\vert A \vert} = p a prime number, then it is the cyclic group p\mathbb{Z}_p.

Proposition

If AA is a finite abelian group and pp \in \mathbb{N} is a prime number that divides the order |A|{\vert A \vert}, then equivalently

This is Cauchy's theorem restricted to abelian groups.

Proof

We prodeed by induction on the order of AA. For |A|=2{\vert A \vert} = 2 we have that A= 2A = \mathbb{Z}_2 is the unique group of order 2 and the statement holds for p=2p =2.

Assume then that the statement has been show for groups of order <n\lt n and let |A|=n{\vert A \vert} = n.

If AA has no non-trivial proper subgroup then nn must be prime and A= nA = \mathbb{Z}_n a cyclic group and the statement follows.

If AA does have a non-trivial proper subgroup HAH \hookrightarrow A then pp divides either |H|{\vert H \vert} or |A/H|\vert A/H\vert.

In the first case by induction assumption HH has an element of order pp which is therefore also an element of GG of order pp.

In the second case there is by induction assumption an element aAa \in A such that a+HA/Ha + H \in A/H has order pp. Since the order of a+HA/Ha + H \in A/H divides the order of aAa \in A it follows that aa has order kpk p for some kk \in \mathbb{N}. Then kak a has order pp.

Theorem

Every finite abelian group is the direct sum of cyclic groups of order p kp^k for a prime number pp \in \mathbb{N} (its p-primary groups).

See for instance (Sullivan).

References

  • John Sullivan, Classification of finite abelian groups (pdf)

Revised on November 17, 2013 23:26:12 by Urs Schreiber (89.204.138.155)