# Groups and subgroups

1. Let G be a group containing subgroups H and K such that we can find an element h e H-K an an element k e K - H. Prove that h o k is not a subgroup of H U K. Deduce that H U K is not a subgroup of G.

I have proved that h o k is not in H U K but I dont know how to deduce that H U K is not a subgroup of G.

If $$h \in H \setminus K$$, $$k \in K \setminus H$$, and you have proved that $$hk \notin H \cup K$$, that shows that $$H \cup K$$ fails to satisfy one of the three basic properties required of a group. Which one?

If $$h \in H \setminus K$$, $$k \in K \setminus H$$, and you have proved that $$hk \notin H \cup K$$, that shows that $$H \cup K$$ fails to satisfy one of the three basic properties required of a group. Which one?

Inverse element?

No, guess again...

SammyS
Staff Emeritus
If $$h \in H \setminus K$$, $$k \in K \setminus H$$, and you have proved that $$hk \notin H \cup K$$, that shows that $$H \cup K$$ fails to satisfy one of the three basic properties required of a group. Which one?