By Justin R. Smith

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E. for all g1 , g2 ∈ G f ( g1 g2 ) = f ( g1 ) f ( g2 ) ∈ H The set of all elements g ∈ G such that f ( g) = 1 ∈ H is called the kernel of f , denoted ker f . If f is bijective, it is called an isomorphism and the groups G and H are said to be isomorphic. An isomorphism from a group to itself is called an automorphism. A homomorphism f : G → H has certain elementary properties that we leave as exercises to the reader: 24 3. GROUP THEORY E XERCISES . 1. Show that f (1) = 1 ∈ H 2. Show that f ( g−1 ) = f ( g)−1 3.

R EMARK . Since a does not affect any of the elements that b permutes and vice-versa, we have ab = ba. 2) (i1 , . . , is ) = (is−1 , is ) · · · (i2 , is )(i1 , is ) Recall exercise 5 on page 22, in which the reader showed that ( ab)−1 = . A simple induction shows that b −1 a −1 1 −1 ( a 1 . . a n ) −1 = a − n · · · a1 in any group. 4) (i1 , . . , is ) = ( a1 , b1 ) · · · ( as−1 , bs−1 )( as , bs ) (i1 , . . , is )−1 = ( as , bs )( as−1 , bs−1 ) · · · ( a1 , b1 ) Since every permutation is a product of disjoint cycles, and every cycle can be expressed as a product of transpositions, it follows that every permutation can be written as a product of transpositions.

For instance a −1 = 2 1 1 2 5 3 3 4 4 5 or, if we sort the upper row into ascending order, we get a −1 = 1 2 2 1 3 4 4 5 5 3 and it is easy to see that a−1 a = aa−1 = 1. 1. If n > 0 is an integer, the group Sn is the set of all permutations of the set {1, . . , n} and is called the symmetric group of degree n. Note that this will have n! elements. 2. A group is a set, G, equipped with a two maps µ: G × G → G ι: G → G called, respectively, the multiplication and inversion maps. We write µ( g1 , g2 ) as g1 g2 and ι( g) as g−1 for all g1 , g2 , g ∈ G, and these operations satisfy (1) there exists an element 1 ∈ G such that 1g = g1 = g for all g ∈ G (2) for all g ∈ G, gg−1 = g−1 g = 1 (3) for all g1 , g2 , g3 ∈ G, g1 ( g2 g3 ) = ( g1 g2 ) g3 If the group G is finite, the number of elements of G is denoted | G | and called the order of G.

### Abstract Algebra [draft] by Justin R. Smith

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