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# absalg22 - Arkansas Tech University MATH 4033 Elementary...

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Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 22 Quotient Groups Let’s look closely on the construction of the group (ZZ n , ) . We know that the elements in ZZ n are equivalence classes of the equivalence relation defined on ZZ by a b if and only if n | ( a - b ) . Also, an element of ZZ n is a right coset. Indeed, if 0 k < n then < n > + k = [ k ] = { nq + k : q ZZ } . The operation is defined by [ a ] [ b ] = [ a + b ] or using right cosets ( < n > + a ) ( < n > + b ) = < n > +( a + b ) . Note that the cyclic group < n > is a normal subgroup of ZZ since ZZ is Abelian (See Example 21.4). We are going to use the above ideas to construct new groups where G is a group replacing ZZ and N is a normal subgroup of G playing the role of < n > . More precisely, we have the following theorem. Theorem 22.1 Let N be a normal subgroup of a group G and let G/N be the set of all right cosets of N in G. Define the operation on G/N × G/N by ( Na )( Nb ) = N ( ab ) . Then ( G/N, · ) is a group, called the quotient group of G by N. Proof. · is a well-defined operation We must show that if ( Na 1 , Nb 1 ) = ( Na 2 , Nb 2 ) then N ( a 1 b 1 ) = N ( a 2 b 2 ) . To see this, since ( Na 1 , Nb 1 ) = ( Na 2 , Nb 2 ) then Na 1 = Na 2 and Nb 1 = Nb 2 . Since a 1 = ea 1 Na 1 then a 1 Na 2 so that a 1 = na 2 for some n N. Sim- ilarly, b 1 = n 0 b 2 for some n 0 N. Therefore, a 1 b 1 = na 2 n 0 b 2 . Since N / G then a 2 n 0 a - 1 2 N, say a 2 n 0 a - 1 2 = n 00 N. Hence, a 2 n 0 = n 00 a 2 so that a 1 b 1 = nn 00 a 2 b 2 . But nn 00 N so that a 1 b 1 N ( a 2 b 2 ) . Since N ( a 2 b 2 ) is an equivalence class and a 1 b 1 N ( a 2 b 2 ) then N ( a 1 b 1 ) = N ( a 2 b 2 ) (Theorem 9.2). · is associative Let a, b, c G. Then Na ( NbNc ) = Na ( Nbc ) = N ( a ( bc )) = N (( ab ) c ) = N ( ab ) Nc = ( NaNb )( Nc ) 1

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where we used the fact that multiplication in G is associative.
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absalg22 - Arkansas Tech University MATH 4033 Elementary...

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