Question:Isomorphic normal subgroups isomorphic quotient groups equivalent as subgroups

From Groupprops
Jump to: navigation, search
This question is about normal subgroup, quotient group, isomorphic groups| See more questions about normal subgroup | See more questions about quotient group| See more questions about isomorphic groups
This question has type complete description conjecture

Q: Suppose H and K are normal subgroups of G such that H and K are isomorphic groups and the quotient groups G/H and G/K are isomorphic groups. Is there an automorphism of G sending H to K?

A: Not in general, though it is hard to construct counterexamples accessible at a very elementary level. A discussion of many minimal counterexamples (with various additional restrictions on the nature of the groups) is available at series-equivalent not implies automorphic. The jargon used there is: two subgroups are termed series-equivalent subgroups if they are isomorphic normal subgroups and the quotient groups are isomorphic. Two subgroups are termed automorphic subgroups if there is an automorphism of the group mapping one to the other.

More basic: Question:Normal subgroup quotient group determine whole group