Register to reply

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

by grimster
Tags: invg, monoid, prove
Share this thread:
Apr12-05, 09:24 AM
P: 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?
Phys.Org News Partner Science news on
Wildfires and other burns play bigger role in climate change, professor finds
SR Labs research to expose BadUSB next week in Vegas
New study advances 'DNA revolution,' tells butterflies' evolutionary history
Apr12-05, 10:10 AM
P: 266
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?
Apr12-05, 12:54 PM
P: 39
this is what i have so far. e is the identity element.

there is an element c such that c a = e.


Apr12-05, 02:53 PM
P: 266
Prove that inv(G) is a group. G is a monoid...

Excuse me- what's the problem with my proof?
Apr12-05, 03:07 PM
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,092
It's not his proof. When doing homework, one should present one's own work, not copy someone else's.
Apr12-05, 03:36 PM
P: 39
Quote Quote by Palindrom
Excuse me- what's the problem with my proof?
i don't know. it looked kind of "sloppy". i guess it is ok, but i wanted something a bit more "structured"...
Apr12-05, 03:56 PM
P: 266
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...

Register to reply

Related Discussions
Isometry sub-group of the gauge group & the center of structural group Differential Geometry 0
Lorentz group, Poincaré group and conformal group Special & General Relativity 12
Solve the algebraic equation,Galois group of FF, and prove/disprove. Calculus & Beyond Homework 6
Prove every even ordered group has an element of order 2 Calculus & Beyond Homework 7
Prove that a set is a monoid, but not a ring. Linear & Abstract Algebra 5