Almost Commutative Property in Groups

[wnorman27]

Homework Statement

I'm trying to figure out if the following property has a name:

for $g\in G$, $h\in H$, $\exists h'\in H$ s.t. gh=h'g.

obviously this is not quite commutativity, but it seems like it might be useful in a variety of situations.

Homework Equations

I've just finished a proof that if a group K has two normal subgroups G and H, whose intersection is just the identity, and whose join is K, then there exists an isomorphism θ(g,h)=gh for all g in G and all h in H. The key to proving surjectivity involved the fact that since H is normal, ghg$^{-1}$ is also in H (call this h') so h=g$^{-1}$h'g and

$gh=gg^{-1}h'g=h'g$

The Attempt at a Solution

I think I've seen this discussed elsewhere, just can't remember the name. ----commutativity?

 

[HallsofIvy]

Homework Statement

I'm trying to figure out if the following property has a name:

for $g\in G$, $h\in H$, $\exists h'\in H$ s.t. gh=h'g.

obviously this is not quite commutativity, but it seems like it might be useful in a variety of situations.

For H a group, the "property" that there exists such an h' doesn't have a name because it is always true! For any such g and h, $h'= ghg^{-1}$ must exist. And that is saying that h and h' are conjugates. Perhaps that is what you are looking for.

Homework Equations

I've just finished a proof that if a group K has two normal subgroups G and H, whose intersection is just the identity, and whose join is K, then there exists an isomorphism θ(g,h)=gh for all g in G and all h in H. The key to proving surjectivity involved the fact that since H is normal, ghg$^{-1}$ is also in H (call this h') so h=g$^{-1}$h'g and

$gh=gg^{-1}h'g=h'g$

The Attempt at a Solution

I think I've seen this discussed elsewhere, just can't remember the name. ----commutativity?

 

[wnorman27]

I understand that in the context of groups, this h' is just the result of conjugation of h by g, but my thought was that perhaps this might occur outside of the context groups (say in cases where inverses may not exist)? I can't think of any examples of this though... maybe this is just not useful.