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Chapter 16: Problem 12

For a group \(G\), prove that the function \(f: G \rightarrow G\) defined by \(f(a)=a^{-1}\) is an isomorphism if and only if \(G\) is abelian.

Short Answer

Expert verified
The function \(f(a) = a^{-1}\) is an isomorphism if and only if \(G\) is an abelian group. The proof has been established in both directions - that if \(G\) is abelian, then \(f\) is an isomorphism, and vice versa. This has been achieved by using the fact that \(f\) is bijective and preserves group operations, and by confirming that \(G\) satisfies the commutative law.

Step by step solution

01

Prove Forward Direction (G is Abelian 鈬� f is an Isomorphism)

Assume that \(G\) is an abelian group. For \(f\) to be an isomorphism, it must be a bijection and must preserve group operation. A function \(f(a) = a^{-1}\) is obviously bijective. Now, for any \(a, b \in G\), \(f(a \cdot b)= (a \cdot b)^{-1}=b^{-1} \cdot a^{-1}\). Since \(G\) is abelian, \(b^{-1} \cdot a^{-1}=a^{-1} \cdot b^{-1} = f(a) \cdot f(b)\). Therefore, \(f\) is an isomorphism.
02

Prove Reverse Direction (f is an Isomorphism 鈬� G is Abelian)

Assume that the function \(f\) is an isomorphism. To prove that \(G\) is abelian, it must be demonstrated that for any \(a, b \in G\),\(a \cdot b = b \cdot a\). As \(f\) is an isomorphism, \(f(a \cdot b) = f(a) \cdot f(b)\). Substituting \(f\) into the equation, \( (a \cdot b)^{-1}=a^{-1} \cdot b^{-1}\). Applying inversion on both sides, \(b \cdot a = a \cdot b\). Hence, \(G\) is an abelian group.

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Most popular questions from this chapter

a) Find all generators of the cyclic groups \(\left(\mathbf{Z}_{12},+\right)\), \(\left(\mathbf{Z}_{16},+\right)\), and \(\left(\mathbf{Z}_{24},+\right)\) b) Let \(G=\langle a\rangle\) with \(o(a)=n\). Prove that \(a^{k}, k \in \mathbf{Z}^{+}\), generates \(G\) if and only if \(k\) and \(n\) are relatively prime. c) If \(G\) is a cyclic group of order \(n\), how many distinct generators does it have?

Why is the set \(\mathbf{Z}\) not a group under subtraction?

Let $$ H=\left[\begin{array}{lllllll} 1 & 1 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 & 1 \end{array}\right] $$ be the parity-check matrix for a Hamming \((7,4)\) code. a) Encode the following messages: \(\begin{array}{llllll}1000 & 1100 & 1011 & 1110 & 1001 & 1111 .\end{array}\) b) Decode the following received words: \(\begin{array}{cccc}1100001 & 1110111 & 0010001 & 0011100\end{array}\) c) Construct a decoding table consisting of the syndromes and coset leaders for this code. d) Use the result in part (c) to decode the received words given in part (b).

If \(G, H\), and \(K\) are groups and \(G=H \times K\), prove that \(G\) contains subgroups that are isomorphic to \(H\) and \(K\).

Express each of the following elements of \(S_{7}\) as a product of disjoint cycles. $$ \begin{aligned} \alpha &=\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 4 & 6 & 7 & 1 & 5 & 3 \end{array}\right) \\ \beta &=\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 6 & 5 & 2 & 1 & 7 & 4 \end{array}\right) \\ \gamma &=\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 3 & 1 & 7 & 5 & 4 & 6 \end{array}\right) \\ \delta &=\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 4 & 2 & 7 & 1 & 3 & 6 & 5 \end{array}\right) \end{aligned} $$
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