Let G a finite group, g E G such
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- Exercise 8 states that every subgroup of an abelian group is normal. Give an example of a nonabelian group for which every subgroup is normal. Exercise 8: Show that every subgroup of an abelian group is normal.Show that the inner direct product group is isomorphic to external direct sum group by constructing a function f:H1 x H2 x ... x Hn ---> H1 + H2 + ... + Hn via h1h2...hn ---> (h1, h2,...,hn), showing it is isomorphism. Please explain each step clearly. Thanks.Write under isomorphism all abelian and non-abelian groups of order 8 and their respective subgroups. Please be as clear as possible and legible. Thank you.
- Give an example of a cyclic group of smallest order that containsboth a subgroup isomorphic to Z12 and a subgroup isomorphic toZ20. No need to prove anything, but explain your reasoning.I need help proving that subgroups and quotient groups of a solveable group are solveable for abstarxct algebraWrite under isomorphism all the abelian groups of order 8 and their respective subgroups. Please show all the steps as clear as possible expleining them. Thank you
- Find Aut(Z15) . Use the Fundamental Theorem of Abelian Groups to express this group as an external direct product of cyclic groups of prime power order. Please be clear with theorems, rules. Be legible.Give an example of an infinite non-Abelian group that has exactlysix elements of finite order.Give an example of a non-Abelian group that has exactly threeelementsof finite order.
- Give an example of the dihedral group of smallest order that contains a subgroup isomorphic to Z12 and a subgroup isomorphic to Z20. No need to prove anything, but explain your reasoning.does the set of polynomials with real coefficients of degree 5 specify a group under the addition of polynomials?Prove that if H is a subgroup of a finite group G of index p, where p is the smallest prime that divides the order of G, then H is normal in G. Please be as clear as possible, showing all the steps and use definitions if necessary. Thank you.