Let G be an abelian group. Let N={ x in G such that x^13=e} 1.) Prove xy is in N
ID: 2986078 • Letter: L
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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