Question

Let G be an abelian group with identity element 'e' an element of G and let n be a positive integer.

1)Show that the subset Gn = { g an element of G: g^n = 'e' } is a subgroup of G

2) Find an example of a group G and an integer n such that Gn is not a subgroup of G

Answer 1

ok first perhaps we have to convince ourselves that in abelian groups the usual laws of exponents hold

Answer 2

what you did before @satellite73 where you just showed that inverse was there was that sufficient enough for this question?

Answer 3

and yeah i learnt that it does hold

Answer 4

that is, for example \((ab)^n=a^nb^n\) but that is because everything commutes

Answer 5

yeahh agreed

Answer 6

so we can show this in one of two ways:
1) show that it is non empty and show that if
\(a,b\in G_n\) then \((ab^{-1})^n\in G_n\) you can look at the wiki page for that one
or else show the following two things,
2) \(e\in G_n\) and
if \(g\in G_n\) then \(g^{-1}\in G_n\) i.e. it is closed under inverses

Answer 7

lets do number 2
a) \(e\in G_n\) because \(e^n=e\)
b) let \(g\in G_n\) and consider \(g^{-1}\in G\) here we must show \(g^{-1}\in G_n\) i.e. explicitly show \((g^{-1})^n=e\)
but because G is abelian, we know \((g^{-1})^n=g^{-n}\) and it is always the case that
\(g^ng^{-n}=e\)
since \(g^n=e\) we know \(g^ng^{-n}=eg^{-n}=g^{-n}=e\) proving that \((g^{-1})^n=e\) i.e. \(g^{-1}\in G_n\) as needed

Answer 8

yeah okay you just went more detail in what you posted on the other post perfectt makes much more sense when you can see al the steps

Answer 9

you know for number 2 when mahmit said n = 0 i didnt know what our G substet would be he said it would be e but not including and got confused what he meant by that

Answer 10

i guess i took some things for granted before, so it is good to flesh them out

Answer 11

as for the last problem, i am fairly sure that \(S_4\) has 9 elements of order 2. since the order of \(S_4\) is 24, they cannot form a subgroup
you should check this

Answer 12

lol S4 ? i am not familiar with that notation

Answer 13

what kind of groups do you know of?

Answer 14

symmetric group on 4 symbols

Answer 15

i just started with this course i learnt how to apply the cayley table groups under multiplication, addition, abelian groups primitive roots

Answer 16

hmm well then i am not sure. it is clear from problem one that you have to pick a non-abelian group, and that was a small one.

Answer 17

true i will search that one up and see what elements does it contain, but thanks a lot i really appreciate it

Answer 18

yw