Let G be an abelian group. Let N={ x in G such that x^13=e} 1.) Prove xy is in N

Question

Let G be an abelian group. Let N={ x in G such that x^13=e}

1.) Prove xy is in N for all x,y in N.

2.) Prove (xy)z=x(yz) for all x,y,z in N.

3.) prove e in N, where e is the identity element of G.

4.) For every x in N, prove x^-1 in N, in which x^-1 is the inverse of x in G.

5.) Is N a group under the operation of G? If so, is it abelian?

Hint: By the construction of N, every element a in N satisfies a^13=e. Conversely, in order to show that an element a in G is in N, it suffices to show a^13=e. Since G is abelian, we have (ab)^n=a^nb^n for all a, b in G and all n in Z.

Explanation / Answer

1).

xy is in N if xy is in G and (xy)^13=e

now, since G is a group then if x,y are in G this implies xy in in G.

also if x and y are in N then

x^13=e nad y^13=e

hence (xy)^13=(x^13)*(y^13)=e*e=e

hence both satisfied so xy is in N

2.)

x,y,z are in N this implies they are in G since N is inG

since G is a group

and hence it is associative.

hence (xy)z=x(yz)

3.)

now, G is an abelian group hence e is in G

now, for e to lie in N it should satisfy e^13=e which is true

hence e lies in N

4.)

x lies in N

x^(-1) is the inverse of x in G

let y=(x^(-1))

now since y is the inverse of x hence xy=e

now x is in N, hence (x^13)=e

power both sides to 13

(x(^13))*(y^13))=(e^13)

e*(y^13)=e

this implies y^13=e

hence y lies in N

hence y=x^(-1) lies in N

inverse lies in N

5.)

using the above four parts we can say that N is a group under the operation of G.

since,

it is associative

has identity element

inverse exists

subgroup of an abelian group G is abelian hence N is abelian.

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