(a) Prove that: If G is a group, a,b € G, |a|=5 and a^3b-ba^3, then ab-ba
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- Find two groups of order 6 that are not isomorphic.Find all homomorphic images of the quaternion group.1.Prove part of Theorem . Theorem 3.4: Properties of Group Elements Let be a group with respect to a binary operation that is written as multiplication. The identity element in is unique. For each, the inverse in is unique. For each . Reverse order law: For any and in ,. Cancellation laws: If and are in , then either of the equations or implies that .
- Prove part e of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.Find all subgroups of the quaternion group.10. Prove that in Theorem , the solutions to the equations and are actually unique. Theorem 3.5: Equivalent Conditions for a Group Let be a nonempty set that is closed under an associative binary operation called multiplication. Then is a group if and only if the equations and have solutions and in for all choices of and in .
- Show that every subgroup of an abelian group is normal.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.Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.
- 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.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.does the set of polynomials with real coefficients of degree 5 specify a group under the addition of polynomials?