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Consequences of equality among left and right cosets


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.6

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 $g \in N_G(H)$.


Solution: 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 $Hg = gH$. Therefore, $g^{-1}Hg = H$ so that $g \in N_G(H)$.


Linearity

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