18.2k views
0 votes
Use induction to prove that ( left(a }right){n}=a b{-n} a ) for all ( a, b \in G ) for ( n geq 0 ).

1 Answer

0 votes

Final Answer:
The statement \((a^n)^(-1) = a^(-n)\) for all \(a, b \in G\) and \(n \geq 0\) can be proven using mathematical induction.

Step-by-step explanation:
To prove this statement by induction, we start with the base case where \(n = 0\). In this case, \((a^0)^(-1) = a^(-0) = e\), where \(e\) is the identity element in the group \(G\). The base case holds true.


Next, assume that the statement holds for some arbitrary \(k \geq 0\), i.e., \((a^k)^(-1) = a^(-k)\). Now, we want to prove it for \(n = k + 1\).


Consider \((a^(k+1))^(-1)\). Using the assumption, we can express \(a^(k+1)\) as \(a \cdot a^k\). Therefore, \((a^(k+1))^(-1) = (a \cdot a^k)^(-1) = (a^k)^(-1) \cdot a^(-1)\). By the induction hypothesis, this is equal to \(a^(-k) \cdot a^(-1) = a^(-(k+1))\).

This completes the inductive step, and by the principle of mathematical induction, the statement
\((a^n)^(-1) = a^(-n)\) holds for all
\(a, b \in G\) and
\(n \geq 0\).Inductive proofs are powerful tools in mathematics, systematically establishing the validity of statements for an infinite set of values based on their truth for specific cases.

User Elvie
by
8.4k points