By Derek J. S. Robinson

This undergraduate textbook for a two-semester path in summary algebra lightly introduces the primary constructions of contemporary algebra. Robinson (University of Illinois) defines the thoughts at the back of units, teams, subgroups, teams performing on units, jewelry, vector areas, box concept, and Galois idea

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7. Let G consist of the permutations (1 2)(3 4), (1 3)(2 4), (1 4)(2 3), together with the identity permutation (1)(2)(3)(4). Show that G is a group with exactly four elements in which each element is its own inverse. (This group is called the Klein3 4-group). 8. Prove that the group Sn is abelian if and only if n ≤ 2. 9. Prove that the group GLn (R) is abelian if and only if n = 1. 3 Groups and subgroups From now on we shall concentrate on groups, and we start by improving the notation. In the first place it is customary not to distinguish between a group (G, ∗) and its underlying set G, provided there is no likelihood of confusion.

7) Every integer n > 1 can be expressed as a product of primes. Moreover the expression is unique up to the order of the factors. 2 Division in the integers 23 Proof. (i) Existence. We show that n is a product of primes, which is certainly true if n = 2. Assume that every integer m satisfying 2 ≤ m < n is a product of primes. If n itself is a prime, there is nothing to prove. Otherwise n = n1 n2 where 1 < ni < n. Then n1 and n2 are both products of primes, whence so is n = n1 n2 . 2). (ii) Uniqueness.

Proof. Consider the crossover diagram on the next page for the transposition (i j ) where i < j . An easy count reveals that the presence of 1 + 2(j − i − 1) crossovers. Since this integer is odd, (i j ) is an odd permutation. The basic properties of the sign function are set out next. 6) Let π, σ ∈ Sn . Then the following hold: (i) sign(π σ ) = sign(π) sign(σ ); (ii) sign(π −1 ) = sign(π). 1 Permutations of a set 1 2 ... i − 1 i i + 1 ... j − 1 j j + 1 ... n 1 2 ... i − 1 j i + 1 ... j − 1 i j + 1 ...