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# if a-b are in G- b^6-e and ab-b^4a- prove b^3-e and ab-baSolutionab-b^.docx

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# if a-b are in G- b^6-e and ab-b^4a- prove b^3-e and ab-baSolutionab-b^.docx

if a,b are in G, b^6=e and ab=b^4a, prove b^3=e and ab=ba
Solution
ab=b^4a

multiply with b^2 on both sides

=>ab^3 = b^6 a

=> b^3 = b^6

given b^6 = e

=> b^3 = e

ab = b^4 a

mutiply and divide by b^3

= b ^7 a / b^3

= b^6 (ba) / b^3

= e (ba) / e

= ba

.

if a,b are in G, b^6=e and ab=b^4a, prove b^3=e and ab=ba
Solution
ab=b^4a

multiply with b^2 on both sides

=>ab^3 = b^6 a

=> b^3 = b^6

given b^6 = e

=> b^3 = e

ab = b^4 a

mutiply and divide by b^3

= b ^7 a / b^3

= b^6 (ba) / b^3

= e (ba) / e

= ba

.

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### if a-b are in G- b^6-e and ab-b^4a- prove b^3-e and ab-baSolutionab-b^.docx

1. 1. if a,b are in G, b^6=e and ab=b^4a, prove b^3=e and ab=ba Solution ab=b^4a multiply with b^2 on both sides =>ab^3 = b^6 a => b^3 = b^6 given b^6 = e => b^3 = e ab = b^4 a mutiply and divide by b^3
2. 2. = b ^7 a / b^3 = b^6 (ba) / b^3 = e (ba) / e = ba