Let G be a group. Let a,b,c denote elements of G, and let e be the nov element of G. 1. Prove that if ab=e, then ba=c (Hint: See theorem 2. 2. Prove that if abc=e, then cab=e and bca = e. 3. State a generalization of pants 1 and 2.

Jerold

Jerold

Answered question

2020-12-14

Let G be a group. Let a,b,c denote elements of G, and let e be the nov element of G.
1. Prove that if ab=e, then ba=c (Hint: See theorem 2.
2. Prove that if abc=e, then cab=e and bca = e.
3. State a generalization of pants 1 and 2.

Answer & Explanation

coffentw

coffentw

Skilled2020-12-15Added 103 answers

Let ab=e. Multiplying from the left by a1 yields a1ab=a1eeb=a1b=a1
Therefore, ba=a1a=e, as required.
2,We can use 1.: abc=e(ab)c=ec(ab)=ecab=e and abc=ea(bc)=e(bc)a=ebca=e
3.If x1,x2xn are such that x1x2xn=e, then xkxk+1xnx1xk1=e

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