We are given whether the question is true or false, but I need to list reasons w

Question

We are given whether the question is true or false, but I need to list reasons why.

1.If [a] = [b] in I_m, then a = b in Z. False
2.There is a homomorphism I_m --> defined by [a] |-->a. False
3.IF a = b in Z, then [a] = [b] in I_m. True
4.If G is a group and K ? G, then there is a homomorphism G --> G/K having kernel K. True
5.If G is a group and K? G, then every homomorphism G --> G/K has kernel K. False
6.Every quotient group of an abelian group is abelian. true
7.If G and H are abelian groups then G x H is an abelian group. True
8.If G and H are cyclic groups, then G x H is a cyclic group. False
9.If every subgroup of a group G is a normal subgroup, then G is abelian. False
10.If G is a group, then {1} ? G and G/{1} ? G True

Explanation / Answer

6.The quotient group is the group of cosets of a subgroup H of abelian group G, with: (aH)(bH) = (ab)H But then, the result is obvious. We have ab = ba, because G is abelian, so trivially: (aH)(bH) = (ab)H = (ba)H = (bH)(aH) 7.cyclic groups states that every subgroup of a cyclic group is cyclic. G and H are isomorphic to Gx{eH} and {eG}xH respECTIVELY 8. It is not even true that if every subgroup of G is normal then G must be abelian 9.It is not even true that if every subgroup of G is normal then G must be abelian. The smallest example is the quaternion group of order 8, Q8={

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