## Consequences of equality among left and right cosets

Let $G$ be a group, $H \leq G$ a subgroup, and $g \in G$ a fixed element. Prove that if $Hg = kH$ for some element $k \in G$, then $Hg = gH$ and $g \in N_G(H)$.

Suppose $Hg = kH$ for some $k \in G$. In particular, $k \cdot 1 \in Hg$; then $k = hg$ for some $h \in H$, so that $Hg = kH= hgH$. Left multiplying by $h^{-1}$ we have $h^{-1}Hg = gH$, but since $h \in H$, $h^{-1}H = H$, and we have $Hg = gH$. Moreover, $g^{-1}Hg = H$ so that $g \in N_G(H)$.

• Bobby Brown  On September 23, 2010 at 8:46 pm

Why is k an element of Hg? I agree that if kH=gH, then k is an element of gH (or for right cosets), but why is this true for “mixing” left and right cosets?

• nbloomf  On September 23, 2010 at 9:05 pm

What is a coset?

$kH$ is the set of all products of the form $kh$ where $h \in H$. Now let $h = 1$.