# Can an infinite conjugacy class in a group split into more than one conjugacy class in some subgroup of finite index?

Are there some theorems about the splitting of infinite conjugacy classes into several conjugacy classes in a subgroup? I am mainly interested in subgroups of finite index. Thanks.

• Take any conjugacy class in any group that splits into more than one class in a subgroup (of finite index if you like). And now just take the product of the entire situation with a group with an infinite conjugacy class? What am I missing? – Kevin Buzzard Nov 20 '09 at 19:01
• But in general, is there a conjugacy class splits into more than one in some subgroup? In abelian group, there is no such conjugacy class. Actually, I want to know whether there some theorems related this. – yeshengkui Nov 21 '09 at 1:41
• "in general is there a conj class that splits into more than one in some subgroup?". In general, yes. – Kevin Buzzard Nov 21 '09 at 7:14

## 1 Answer

I'm not sure what you mean by an infinite length conjugacy class. Most likely you mean that the cardinality is infinite.

Consider the Heisenberg group generated by 3 elements $x,y$ and $z$ with relations so that $z$ is central and $xyx^{-1}y^{-1}=z$. Then the conjugacy class containing $y$ consists of all elements of the form $yz^n$ for integers $n$.

If we pass to the finite index subgroup generated by $x^k,y$ and $z$ for some natural number $k$ this splits into $k$ distinct classes represented by $yz^i$ for $i=0,\ldots k-1$.

• Perhaps I should add that if you pass to a subgroup of index n each class can split into at most n new classes and n can occur as illustrated in my answer. I suppose an orbit-stabiliser type argument will show the number of classes must divide n. – Simon Wadsley Nov 20 '09 at 22:49
• This is a good example! – yeshengkui Nov 21 '09 at 1:37