Suppose G is solvable,and π(G)= {2,m,n}, O_{2}(G)=1.Then can we use the solvability of G proof that O_{2^{\prime}}(G) \neq 1?Let \bar{G}= G / O_{2^{\prime}}(G),what about O_{2}(\bar{G})? I think O_{2}(\bar{G})=\bar{R} ,R is the Sylow 2-subgroup of G.Is this right?What is the specific proof?
The intersection of product of Sylow subgroups
Bob
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