If(G, *) is an abelian group, then for all a, b € G, prove that by mathematical induction -

Mumbai University > Computer Engineering > Sem 3 > Discrete Structures

Marks: 5 Marks

Year: Dec 2013

$(a*b)^n=a^n*b^n$

1. n=0. we have $(ab)^0=e \hspace{5cm} \text{[by definition}]$

   Also $a^0b^0=ee=e.\\ \therefore (ab)^0=a^0b^0.$

2. n > 0. If n=1, then $(ab)^1=ab=a^1b^1.$

   Now suppose for n=k, $(ab)^k=a^kb^k$

   Then $(a+b)^{k+1}$ $=(ab)^k(ab)=a^kb^kab=a^kab^kb \hspace{3cm} [\because \text{G is abelian} \Rightarrow b^ka=ab^k] \\ =a^{k+1}b^{k+1}$

   Thus the result is ture for n=k+1 if it was ture for n=k. But …