As a ZZ-module, the p-primary components of a finite abelian group are its Sylow subgroups

Let M be a finite abelian group of order a = \prod p_i^{k_i} and consider M as a \mathbb{Z}-module in the natural way. Prove that M is annihilated by (a), that the p_i-primary component of M is the (unique) Sylow p_i-subgroup of M, and that M is isomorphic to the direct product of its Sylow subgroups.

Certainly for all m \in M, a \cdot m = 0. Thus (a) \subseteq \mathsf{Ann}_\mathbb{Z}(M). Since \mathbb{Z} is a principal ideal domain, by this previous exercise, M = \bigoplus (q_t)M, where q_t = \prod_{i \neq t} p_i^{k_i}.

We claim that each (q_t)M is a Sylow subgroup of M. To that end, note that p_t^{k_t} \cdot (q_t)M = 0; in particular, every element of (q_t)M has p_t-power order. Conversely, if m \in M has p_t-power order, then m \in \mathsf{Ann}_M(p_t^{k_t}) = (q_t)M. Hence the (q_t)M, that is, the p_t-primary components, are Sylow subgroups of M. Since M is abelian, there is a unique Sylow p_t-subgroup for each p_t; thus M = \bigoplus (q_t)M is the internal direct sum of its Sylow subgroups. Since the direct sum is finite, M is in fact the internal direct product of its Sylow subgroups.

Post a comment or leave a trackback: Trackback URL.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: