Is a group of order p 2 abelian?

Is a group of order p 2 abelian?

(a) A group of order p2 is abelian. Then the center must have order p and it follows that the order of the quotient G/Z(G) is p, hence G/Z(G) is a cyclic group.

Why a group of order p 2 is abelian?

Solution: Let P be a group of order p2. We proved that every p−group has non-trivial, so the center Z(P) of P is not trivial. Thus |Z(P)| is either p or p2. In the latter case, we have P = Z(P), so P is abelian.

Is P group abelian?

Nor need a p-group be abelian; the dihedral group Dih4 of order 8 is a non-abelian 2-group. However, every group of order p2 is abelian.

Is every group of order p 3 abelian?

From the cyclic decomposition of finite abelian groups, there are three abelian groups of order p3 up to isomorphism: Z/(p3), Z/(p2) × Z/(p), and Z/(p) × Z/(p) × Z/(p). These are nonisomorphic since they have different maximal orders for their elements: p3, p2, and p respectively.

Which order group is always abelian?

All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic.

Is every group of order 4 abelian?

This implies that our assumption that G is not an abelian group ( or G is not commutative ) is wrong. Therefore, we can conclude that every group G of order 4 must be an abelian group.

Is every group of prime order abelian?

Thus, every group of prime order is cyclic. So, G is abelian. Thus, every cyclic group is abelian.

How do you prove abelian group?

Ways to Show a Group is Abelian

1. Show the commutator [x,y]=xyx−1y−1 [ x , y ] = x y x − 1 y − 1 of two arbitary elements x,y∈G x , y ∈ G must be the identity.
2. Show the group is isomorphic to a direct product of two abelian (sub)groups.

Are all p groups cyclic?

For a prime number p, the group (Z/pZ)× is always cyclic, consisting of the non-zero elements of the finite field of order p. More generally, every finite subgroup of the multiplicative group of any field is cyclic.

Is group 125 abelian order?

|G|=125 is possible since for any prime p there exist a non-abelian group of order p3. In a group of order 35 the 7 group and the 5 group are normal by the Sylow theorems and hence this group is cyclic.

Which is abelian group?

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative.

Is group of order 9 abelian?

Proof: Let G be a group of order 9. If G contains an element of order 9 then it is cyclic and hence abelian, so we must consider the case when every element has order 3 in the group.