please give a rigorous proof of the question in the title, every part of the que

Question

please give a rigorous proof of the question in the title, every part of the question has its own post. I will give A+ ratings to detailed answers which explain as well. Thank you in advance

Explanation / Answer

index of subgroup Y in X is o(X)/o(Y). So, if K is a subgroup of H then -> index of K in G = o(G)/o(K). index of H in G = o(G)/o(H). index of K in H = o(H)/o(K). Then o(G)/o(H)*o(H)/o(K) = o(G)/o(K). so, we have to show that K is a subgroup of H. So, K is contained in H. To, show that K is a subgroup of H, we need to show that K is closed that is enough. The rest of the properties come directly from the fact that K is a subgroup of G. So, k1, k2 in K, we need to show that k1k2 in K this is true because K subgroup G => k1k2 in K. The other properties like identity, inverse and associativity are also implied because K is subgroup G. Hence proved Hence done. Message me if you have any doubt

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.