Theorem: If $H$ is a normal subgroup of $G$, then $aH=Ha$ for every $a \in G$.
If H a subgroup, and the left and right cosets are equal, does this mean that H is the center? Isn't the definition of the center: the set of all a in G s.t. ax=xa for all x in G. But if the left and right cosets are equal isn't that the definition of center?
I know that the center of a group is normal, is every normal subgroup of a group the center of the group?
Best Answer
The equality $aH=Ha$ means that the sets $$aH=\{ah:h\in H\}$$ and $$Ha=\{ha:h\in H\}$$ are equal.
But if this is the case and if you pick some $h\in H$, you only know that $ah\in Ha$. That is, there exists some $h'$ in $H$ such that $ah=h'a$, and $h$ and $h'$ may be different.
Furthermore, if it is the case that $$\forall h\in H(ah=ha)$$ that means that every element of $H$ commutes with every element of $G$. This means that $H$ is included in the center. The center of a group $G$ is unique, and it is the set of elements that commute with every element of $G$. (Exercise: can you show that this set is indeed a subgroup?)