Let G be a group with identity e. Let H be an abelian non-trivial proper subgroup of G with the property that H ∩ gHg−1 = {e} for all g / ∉ H.
If K = \(\{g \in G: g h=h g \text { for all } h \in H\}\), then
1
K is a proper subgroup of H
2
H is a proper subgroup of K
3
K = H
4
there exists no abelian subgroup L ⊆ G such that K is a proper subgroup of L