Question

Give an example of each object described below, or explain why no such object exists:

1. A group with 11 elements that is not cyclic.

2. A nontrivial group homomorphism f : D8 −→ GL2(R).

3. A group and a subgroup that is not normal.

4. A finite integral domain that is not a field.

5. A subgroup of S4 that has six elements.

Answer 1

1.the statement is false.

Because, we know that, every group of prime order is cyclic. Since, 11 is cyclic hence it is always cyclic group. So there exist no group of order 11 that is not cyclic.

3.)yes, such a example exists.

Consider the Symmtric group S3 whose order is 6. It has a subgroup of 2, namely H={e,a}. But it is not normal.

4) No, such a example does not exist.

Because, every finite integral domain is always field.

5).No.such a result is not exist. Because S4 has no subgroup of order 6.

Because converge of Lagranges theorem is not true. This is true only when the group is finite abelian group. But S3 is non - abelian.