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Chapter 16: Problem 5

Define the binary operation \(\circ\) on \(\mathbf{Z}\) by \(x \circ y=x+y+1\). Verify that \((\mathbf{Z}, \circ)\) is an abelian group.

Short Answer

Expert verified
Yes, \((\mathbf{Z}, \circ)\) where \(x \circ y=x+y+1\) is an abelian group because it satisfies all the necessary group properties: closure, identity, associativity, existence of inverse, and commutativity.

Step by step solution

01

Verify Closure

Closure under an operation means that the result of the operation, applied to any two elements in the set, is always an element in the set. So, we first check if \(x \circ y = x+y+1\) is an integer when both x, y are integers. Since the sum and addition of any two integers is an integer, \(x \circ y\) is indeed an integer. Therefore, the operation is closed.
02

Identity Element

-1 is the identity element of this binary operation. For any integer \(x\), \(x \circ (-1) = x+(-1)+1=x\) and \(-1 \circ x = -1+x+1=x\). Hence, -1 is the identity.
03

Associativity

We verify associativity, which states that for all \(x, y, z\) in \(\mathbf{Z}\), \((x \circ y) \circ z = x \circ (y \circ z)\). This computation gives us \((x+y+1)+z+1=x+ (y+z+1)+1\), which implies the operation is associative.
04

Inverse Element

Every element in the group should have an inverse. For any integer \(x\), the inverse element would be \(-x-2\), because \(-x-2 \circ x = -x - 2 + x + 1 = -1\), which is the identity element, and similarly \(x \circ (-x - 2) = -1\). Hence, the inverse of any integer \(x\) under this operation is \(-x - 2\).
05

Commutativity

Finally, we check the operation for commutativity. This implies \(x \circ y = y \circ x\). For every \(x, y\) in \(\mathbf{Z}\), \(x+y+1=y+x+1\). So, the operation is commutative.

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Most popular questions from this chapter

a) Consider the group \(\left(\mathbf{Z}_{2} \times \mathbf{Z}_{2}, \oplus\right)\) where, for \(a, b, c, d \in \mathbf{Z}_{2},(a, b) \oplus(c, d)=(a+c, b+d)-\) the sums \(a+c\) and \(b+d\) are computed using addition modulo 2. What is the value of \((1,0) \oplus(0,1) \oplus(1,1)\) in this group? b) Now consider the group \(\left(\mathbf{Z}_{2} \times \mathbf{Z}_{2} \times \mathbf{Z}_{2}, \oplus\right)\) where \((a, b, c) \oplus(d, e, f)=(a+d, b+e, c+f)\). (Here the sums \(a+d, b+e, c+f\) are computed using addition modulo 2.) What do we obtain when we add the seven nonzero (or nonidentity) elements of this group? c) State and prove a generalization that includes the results in parts (a) and (b).

Let \(H\) and \(K\) be subgroups of a group \(G\), where \(e\) is the identity of \(G\). a) Prove that if \(|H|=10\) and \(|K|=21\), then \(H \cap K=\\{e\\}\). b) If \(|H|=m\) and \(|K|=n\), with \(\operatorname{gcd}(m, n)=1\), prove that \(H \cap K=\\{e\\}\).

Let \(f: G \rightarrow H\) be a group homomorphism onto \(H\). If \(G\) is abelian, prove that \(H\) is abelian.

If \(G\) is a group of order \(n\) and \(a \in G\), prove that \(a^{n}=e\).

Let \(R\) be a ring with unity \(u\). Prove that the units of \(R\) form a group under the multiplication of the ring.
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