Let G be a group with |G|=4. Define the map Λ: G→Sym(G) by Λ(g)(x)=g o x for all g in the group G and all x in the set G. Prove: Either G is cyclic or G is isomorphic to V₄.
Let G be a group with |G|=4. Define the map Λ: G→Sym(G) by Λ(g)(x)=g o x for all g in the group G and all x in the set G. Prove: Either G is cyclic or G is isomorphic to V₄.
your map, $\displaystyle \Lambda,$ is injective. so $\displaystyle G$ can be considered as a subgroup of $\displaystyle \text{Sym}(G) \cong S_4.$ now what are the subgroups of order $\displaystyle 4$ in $\displaystyle S_4$?
Last edited by NonCommAlg; Nov 30th 2009 at 03:30 AM.