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Extra resources for Abstract Algebra [draft]

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The subgroup of inner automorphisms is a normal subgroup of the group of all automorphisms. The quotient is called the group of outer automorphisms. Out( G ) = Aut( G ) Inn( G ) 18. If G is a group, show that the commutator subgroup, [ G, G ], is a normal subgroup. 19. If G is a group, show that the quotient G [ G, G ] is abelian. This is often called the abelianization of G. 5. , a subgroup of a symmetric group. 1 (Cayley). If G is a group, there exists a symmetric group SG and an injective homomorphism f : G → SG where SG is the group of all possible permutations of the elements of G.

If G is a group with a subgroup H ⊂ G and g ∈ G, then the conjugate of H by g is defined to be g · H · g−1 , and denoted H g . A subgroup H ⊂ G of a group is said to be normal if H = H g for all g ∈ G. This fact is represented by the notation H G. R EMARK . For H to be a normal subgroup of G, we do not require ghg−1 = h for h ∈ H and every g ∈ G — we only require ghg−1 ∈ H whenever h ∈ H. If G is abelian, all of its subgroups are normal because ghg−1 = gg−1 h = h. Here’s an example of a non-normal subgroup: Let S= 1, a = 1 2 2 1 3 3 g= 1 1 2 3 3 2 and let ⊂ S3 so that g2 = 1 which means g−1 = g.

Gn , this is a subgroup of G, called the subgroup generated by { g1 , . . , gn }. If G = g1 , . . , gn , we say that the group G is generated by g1 , . . , gn . A group that is generated by a single element is called cyclic. 10. If G is a group and g ∈ G is an element, the set of all powers of g forms a subgroup of G denoted g ⊂ G. If G is finite, the order of this subgroup is called the order of g, denoted ord( g). 11. If G and H are groups, their direct sum, G ⊕ H, is the set of all possible pairs ( g, h) with g ∈ G, h ∈ H and group-operation ( g1 , h1 )( g2 , h2 ) = ( g1 g2 , h1 h2 ) The group is called the Klein 4-group.

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