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## Main Question or Discussion Point

Hi,

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

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

are

Moreover we have that:

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

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

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?

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 G2) [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?