What is the difference between direct sum and direct product between groups $\mathbb{Z}_n$ and $\mathbb{Z}_m$?

I that the same? I found in different literature different notation and I'm very confused about that.

  • $\begingroup$ Some authors adhere to the convention that we say direct sum if both groups are abelian, direct product otherwise. $\endgroup$ Jun 22, 2015 at 13:25

1 Answer 1


For only finitely terms of $G_i, i\in I$, where $I=1,...n$, the direct product and direct sum is the same $\prod_{i\in I} G_i=\sum_{i\in I} G_i$.

However, if $I$ is a infinite set, $\prod_{i\in I} G_i$ is much bigger than $\sum_{i\in I} G_i$ in the following sense: $\sum_{i\in I} G_i$ consist of element $g=\{g_i\}_{i\in I}$ and $g_i \in G_i$, where $g_i\neq e_i$ for only finitely terms. While $\prod_{i\in I} G_i$ admits element with infinite nonzero terms.

  • $\begingroup$ Oh, ok. Thank You. I was looking Z_2 and Z_3 which are finite so everything makes sense now. :) $\endgroup$
    – user222801
    Jun 22, 2015 at 12:48
  • 1
    $\begingroup$ @user222801 You may accept the answer if it completely solves your problem, just beneath the button for voting up or down. $\endgroup$
    – AG learner
    Jun 23, 2015 at 15:08

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