Math Is Fun Forum

Seetha Rama Raju Sanapala

Member

Registered: 2016-10-23

Posts: 4

What are the minimum requirements of a group?

Regarding group conditions, all books say closure, associativity, same identity (left-side, right-side i.e a*e  = e*a =a) are required.  But about inverse they differ.  Some books say, for every element a,  there should be an element such b that a*b = e where e is identity element.  But some other say that for every element a there should be such that a*b =e.  Without mentioning  that b*a also should be e,.  b*a = e is it not required.  Can it be derived from other properties?

Last edited by Seetha Rama Raju Sanapala (2016-10-25 05:54:28)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: What are the minimum requirements of a group?

hi Seetha Rama Raju Sanapala

I think it can.  Does this work?

Let b be the right inverse of a.

I have left out some associativity steps to make the proof simpler.

(ba)(ba) = b(ab)a = bea = ba

Let c be the right inverse of ba

(ba)(ba)c = (ba)c

(ba)e = e

ba = e

So b is a left inverse of a.

Note:  Every other property of the group is required for this.

Bob

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