1. How many distinct subgroups does the group Z 24 have ? 2. Suppose that G is a

ID: 3116088 • Letter: 1

Question

1. How many distinct subgroups does the group Z24 have ?

2. Suppose that G is a group and N a normal subgroup of G, and H = G/N. Is it always true that G is isomorphic to the product group N × H ?

3. Using the class equation prove that if p is a prime, then every group whose order is a positive power of p has a nontrivial center.

4. Let p be a prime. Prove that every finite abelian group whose order is divisible by p must have a subgroup isomorphic to Zp.

5. Let p be a prime number. Can a group of order p3 be non-abelian ?

1- there are 24 Distinct subgroups does the group Z24 have.

2- yes it will be isomorphic to the product group NxH

3- yes if p is a prime,then every group whose order is a positive power is p has a nontrivial centre.

5- no it can't be a order of p3 be non abelian.

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