# Is there an infinite group that has finite subgroup with finite index?

Is there an infinite group that has finite subgroup with finite index?

I can find one, if the index is infinite - but what if the index is finite?

Thanks

• But... isn't the cardinality of the group the product of cardinality and index of the subgroup? – pjs36 Dec 14 '15 at 21:46
• Better questions would have been: "Is there an infinite group that has an infinite subgroup with finite index?" or "Is there an infinite group that has a finite subgroup with infinite index?" or "Is there an infinite group that has an infinite subgroup with infinite index? – Nicky Hekster Dec 14 '15 at 22:48

No. If $H$ is a finite subgroup of finite index in a group $G$, then $\left| G\right| = \left| H\right|[G:H]$ is finite.