(a) What does it mean for two groups to be isomorphic?
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- Find two groups of order 6 that are not isomorphic.Exercises 35. Prove that any two groups of order are isomorphic.Prove part c 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.
- 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 .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.Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.
- Lagranges Theorem states that the order of a subgroup of a finite group must divide the order of the group. Prove or disprove its converse: if k divides the order of a finite group G, then there must exist a subgroup of G having order k.24. Prove or disprove that every group of order is abelian.Exercises 8. Find an isomorphism from the group in Example of this section to the multiplicative group . Sec. 16. Prove that each of the following sets is a subgroup of , the general linear group of order over .
- Let H be a subgroup of a group G. Prove that gHg1 is a subgroup of G for any gG.We say that gHg1 is a conjugate of H and that H and gHg1 are conjugate subgroups. Prove that H is abelian, then gHg1 is abelian. Prove that if H is cyclic, then gHg1 is cyclic. Prove that H and gHg1 are isomorphic.9. Find all homomorphic images of the octic group.Prove that if is an isomorphism from the group G to the group G, then 1 is an isomorphism from G to G.