# Prove that inv(G) is a group. G is a monoid

1. ### grimster

39
G is a monoid. Inv(G) = {a E G, exists b so that a b = b a = 1}

prove that Inv(G) is a group. it's pretty obvious that inv(G) is a group. a monoid is a set with a law of composition which is associative and has a unit element. so inv(G) is clearly a group, because for all a in inv(G) there is an inverse element. but how do i prove this?

i guess i have to show that for each x in inv(G), the inverse of x is also in G. but how do i do that?

Last edited: Apr 12, 2005
2. ### Palindrom

263
Given x in inv(G), you know there is a y in G so that xy=yx=1.
Let's look at y. Guess what? yx=xy=1. But then y is in inv(G) by definition...
1 is in inv(G), * is associative... did I forget something?

3. ### grimster

39
this is what i have so far. e is the identity element.

there is an element c such that c a = e.
a=ea=cba=cbae=eae=ae
further
cba=ce=c
and
cba=ea=a

have
ab=ba=e

4. ### Palindrom

263
Excuse me- what's the problem with my proof?

5. ### Hurkyl

15,998
Staff Emeritus
It's not his proof. When doing homework, one should present one's own work, not copy someone else's.

6. ### grimster

39
i don't know. it looked kind of "sloppy". :rofl: i guess it is ok, but i wanted something a bit more "structured"...

7. ### Palindrom

263
I agree with you both- but that's why I gave it in a sloppy way. It's not hard to formalize it, but it's your job...