Let G be a group, H⊆G a subgroup and a∈G an element of the group. Is it possible that aHa−1⊂H, but aHa−1≠H?

If H has finite index or finite order, this is not possible.

**Answer**

Consider the group of matrices G={[xy01]:x∈Q×,y∈Q}=AGL(1,Q) and its subgroup H={[1y01]:y∈Z}≅Z and of course the single element a=[2001] A direct calculation gives aHa−1={[12y01]:y∈Z}<H is a proper subgroup of *H*.

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**Attribution***Source : Link , Question Author : Sasha , Answer Author : principal-ideal-domain*