if $A$ and $B$ are infinite groups and

$$ |A|\leq|B|, $$

where $|{}\cdot{}|$ denotes the group cardinality, is it right to say that $|A|\cdot|B| = |B|$?

  • $\begingroup$ How about $A=B=\mathbb{Z}/2\mathbb{Z}$? $\endgroup$ – kccu May 4 at 12:58
  • $\begingroup$ If $B$ has infinite cardinality and $A \neq \emptyset$ (which is true if $A$ is a group), then $|A||B|=\max\{|A|,|B|\}=|B|$ since $|A| \leq |B|$. But this does not hold when $A$ and $B$ are finite. $\endgroup$ – kccu May 4 at 13:06
  • $\begingroup$ I meant inifinite groups. how can I prove this? thanks $\endgroup$ – KIMKES1232 May 4 at 13:30
  • $\begingroup$ If you mean infinite groups, then you should edit the question to say what you mean, please. $\endgroup$ – Gerry Myerson May 4 at 13:32
  • $\begingroup$ @KIMKES1232 It has nothing to do with them being groups, this is true of all sets. See cardinal multiplication. $\endgroup$ – kccu May 4 at 13:41

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