Question

Let (G,∗) be a binary structure that has the following properties:

1) The binary operation ∗ is associative.

2) There exists an element e∈G such that for all a∈G, e∗a=a (Existence of left Identity).

3) For all a∈G, there exists b∈G such that b∗a=e (existence of left inverses)

Prove that (G,∗) is a group.

Answer 1

All you need to show, is that it also has a right identity, and a right inverse.

Answer 2

yea,,, I have hourssss trying to :(

Answer 3

That's true as well for non-Abelian groups, right?

Answer 4

Wait, yes it is. I was derping; I forgot the exact definition of Abelian for a second.

Answer 5

It is true for non-abelian groups too, but I'm starting to see why he was having problems. I keep getting that statements that use circular logic.

Answer 6

I know that in order to prove the right identity i use some how my left inverse...
but i keep getting stuck

Answer 7

Man, I'm too sleepy for this. :(

Answer 8

bad reference... please... i have 3 nights with no sleep :(

Answer 9

I'm running in circles, and I can't tell if I'm just falling for a trap, or if I'm just derping because I'm sleepy.

Answer 10

:(

Answer 11

According to the always helpful wikipedia page http://en.wikipedia.org/wiki/Elementary_group_theory,

Let \(y\) be the inverse of \(a*x\), and \(x\) be the inverse of \(a\). Then \[e=y*(a*x)\]\[e=y*(a*(e*x))\]\[e=y*(a*((x*a)*x))\]\[e=y*(a*(x*(a*x)))\]\[e=y*((a*x)*(a*x))\]\[e=(y*(a*x)*a)*x\]\[e=e*(a*x)\]\[e=a*x\]Thus we have a right inverse also.

Answer 12

As for right identity, let \(x\) be the inverse of \(a\). Then\[a*e=a*(x*a)\]\[a*e=(a*x)*a\]\[a*e=e*a\]\[a*e=a\]So we also have right identities.

Answer 13

That took some time to take apart and figure out exactly what was going on.

Answer 14

im still trying to understand the inverses

Answer 15

Any particular step in that proof?

Answer 16

well,, im trying to understand step by step... let me see if i get stuck

Answer 17

Woops, I tried proving it backwards using the right inverse. I feel dumb.

Answer 18

waittt

Answer 19

if x is the inverse of a and if we have left inverse the will be (x*a) nooo (a*x)

Answer 20

If that was unclear, it means that \(x*a=e\).

Answer 21

And we want to show that \(a*x=e\) as well.

Answer 22

yea... but u are using the right one

Answer 23

yea but you can apply that statement like its true.. u want to get that statement

Answer 24

In the proof, using meticulous use of associativity, it shows that if \(y*(a*x)=e\), and \(x*a=e\), then \(a*x=e\). Both \(y\) and \(x\) are left inverses of \(a*x\) and \(a\) respectively.

Answer 25

but my understanding is that u cant just say x*a=e=a*x.. u have to get the answer

Answer 26

We didn't just say that. By converting \(x\) into \(e*x\) and \(e\) into \(x*a\) (true by hypothesis), we can reorganize our terms to prove that \(x*a=a*x=e\)

Answer 27

ok let em check again

Answer 28

I need to go to bed now, but if you continue to have questions, feel free to ask and I'll get back to you tomorrow.

Answer 29

ok thanks you very much
I kind of getting it..