Thread: Groups

3 problems i am unable to solve. please help me out.

1) If every element of a group G is its own inverse. Prove that G is an abelian

2) Show That the set G={a to the power of n | n belogs to I} is an abelian under +4

3)On q-{1} a set of rational numbers except 1. Define * as a*b = a+b-ab for all a,b belongs to q-{1}. Is this a group . justify

Please help me out.

what have you tried? help is more likely to be forthcoming, if people know exactly what steps are giving you trouble.

1) hint: if every element is its own inverse, this means x^2 = e, for all x in G. play around with (xy)^2 = xyxy.

2) what is I? what exactly is "under +4"?

3) verify the group axioms for this operation. be sure to show that a*b is never 1 (closure).

let a,b in G we want to show that

a*b = b*a

(a*b)*(a*b) = e = a*b*(b*a)

Originally Posted by Amer

(a*b)*(a*b) = e = a*b*(b*a)

amer, how do you justify this statement (particularly the right equality)?

Originally Posted by Deveno

amer, how do you justify this statement (particularly the right equality)?

a*b in G since a,b in G and G is group
and from the property the inverse of the element itself
(a*b)*(a*b) = e
a*b*(b*a) = a*(b*b)*a = (a*e)*a = a*a = e
so the equality holds
am I right ?
( when I finished my post i saw your solution if i saw it earlier i would not post mine )

now you have a justification for your statement. but you are still missing how this proves a*b = b*a (although you don't need much to prove this).

that said, in all fairness to the original poster, we ought not to just "hand" the answer to him. i understand you want to help, and that is commendable.

however, have you ever heard the saying: "give a man a fish, and he eats for a day, teach him how to fish, and he eats for life"?

someone who is just given the answer to their problem, is likely to copy it, consider the proble "solved", and move on. they learn nothing, and

perhaps what is worse, do not acquire the habit of trying to figure out things for themselves.

there is a reason i did not answer the question in its entirety, and that is because i want to see some evidence from the thread-starter he is trying to solve

this problem, and not just using this forum as an unpaid homework service.

Originally Posted by Deveno

now you have a justification for your statement. but you are still missing how this proves a*b = b*a (although you don't need much to prove this).

that said, in all fairness to the original poster, we ought not to just "hand" the answer to him. i understand you want to help, and that is commendable.

however, have you ever heard the saying: "give a man a fish, and he eats for a day, teach him how to fish, and he eats for life"?

someone who is just given the answer to their problem, is likely to copy it, consider the proble "solved", and move on. they learn nothing, and

perhaps what is worse, do not acquire the habit of trying to figure out things for themselves.

there is a reason i did not answer the question in its entirety, and that is because i want to see some evidence from the thread-starter he is trying to solve

this problem, and not just using this forum as an unpaid homework service.

I know how to solve add a*b to both sides from the left you will answer the question
you are right about all what you said