- #1

- 713

- 5

it is known that given two subgroups [itex]H\subset G[/itex], and [itex]K\subset G[/itex] of some group G, then we have that:

1) H, K are

*normal*subgroups of G

2) [itex]H\cap K[/itex] is trivial

are

*sufficient*conditions for H and K to

*commute*.

Moreover we have that:

H, K commute [itex]\Rightarrow[/itex] H, K are normal.

In fact, conditions 1) and 2) together are not

*necessary*conditions for commutativity because there exist subgroups that are commutative but do

*not*have trivial intersection (it is posible to find examples).

My question is: is it possible to keep condition 1) and instead replace only 2) with some weaker condition that would make 1),2) necessary and sufficient conditions for commutativity?