Example of subgroup $H$ with the property $g H g^{-1}$ is properly contained in $H$.

Question

After learning the definition of normal subgroup, I would like to find an example of a group $G$ which has a subgroup $H$ such that there exists an element $g \in G$ such that $gHg^{-1} \subsetneq H$, namely, $gHg^{-1}$ is a proper subgroup of $H$.

Can anyone provide an example? Thank you very much!

Thanks! Great example 1.33! I think you can put it to the answer, so that I can accept it. :) — user119882, Jan 12 '16 at 08:35
This must be a candidate for the most frequently asked question. — Derek Holt, Jan 12 '16 at 10:37

score 2 · Accepted Answer · answered Jan 12 '16 at 08:44

2

See Example 1.33 in J.S. Milne's lecture notes.

answered Jan 12 '16 at 08:44

Giuseppe Negro

32,319

1

Links die, so it is useful to accompany a link with something about what it contains - for example, here just give the example without proof! – user1729 Jan 12 '16 at 12:24
@user1729: You are right. However, I have to say that (IMHO) those notes are among the most longeve links in all mathematical Internet (they started in the '90s). Anyway, I'll try to rewrite the example here or to write a stabler reference if time allows me. – Giuseppe Negro Jan 12 '16 at 16:36

Example of subgroup $H$ with the property $g H g^{-1}$ is properly contained in $H$.

1 Answers1