# cardinality of multiplication of groups A,B when $A\leq B$

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

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

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

• How about $A=B=\mathbb{Z}/2\mathbb{Z}$? – kccu May 4 at 12:58
• 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. – kccu May 4 at 13:06
• I meant inifinite groups. how can I prove this? thanks – KIMKES1232 May 4 at 13:30
• If you mean infinite groups, then you should edit the question to say what you mean, please. – Gerry Myerson May 4 at 13:32
• @KIMKES1232 It has nothing to do with them being groups, this is true of all sets. See cardinal multiplication. – kccu May 4 at 13:41