By F. M. Hall
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Additional resources for An introduction to abstract algebra,
Prove that G is abelian. 4–7 Let G1 and G2 be groups. Prove that G1 × G2 ∼ = G 2 × G1 . 4–8 Let G be a finite group and let x be an element of G. Prove that xn = e for some positive integer n. Hint: Use the fact that the elements x1 , x2 , x3 , . . cannot be distinct. 4–9 Let G be a group of even order. Prove that there exists a nonidentity element x of G such that x2 = e. 4–10 Let ϕ : G → G be an isomorphism from the group G to the group G . 1). Prove that ϕ−1 is an isomorphism. 4–11 Prove that Q is not isomorphic to Z (both groups under addition).
Let m and n be integers with n > 0. There exist unique integers q and r with 0 ≤ r < n such that m = qn + r. We omit the proof and instead appeal to the reader’s knowledge of long division, which is used to divide m by n as follows: 46 q n)m .. r m r =q+ n n ⇒ ⇒ m = qn + r. So the q in the theorem corresponds to the whole part of the quotient and the r corresponds to the remainder. In long division, one continues the algorithm until the remainder is less than the divisor, that is, 0 ≤ r < n (cf.
For x, y ∈ R, we have ϕ(x + y) = ex+y = ex ey = ϕ(x)ϕ(y), so ϕ satisfies the homomorphism property. Therefore, ϕ is an isomorphism and we conclude that R ∼ = R+ . 3 Example Is R isomorphic to Z? 6 that |R| = |Z|, that is, there is no bijection from R to Z. In particular, there cannot be an isomorphism from R to Z. Therefore, R is not isomorphic to Z. 35 If G ∼ = G , then G and G are indistinguishable as groups. If G has a property that can be described just using its elements and its binary operation, then G must have that same property, and vice versa.