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Algebra 2 Monoids and Groups Monoids and groups Normal subgroups Algebra Interactive Algebra 2 Monoids and groups Normal subgroups A.M. Cohen, H. Cuypers, H. Sterk A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 1 / 25 Monoids and groups Normal subgroups Algebra Interactive In general, left cosets need not coincide with right cosets. If they do, we have a case that deserves special attention. Let G be a group. A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 2 / 25 Monoids and groups Normal subgroups Algebra Interactive Theorem Normality Let H be a subgroup of G. The following asser- tions are equivalent. 1 g·H=H·g for every g∈G. −1 2 g·h·g ∈H for every g, h∈G. If H satisfies these properties, it is called a normal subgroup of G. A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 3 / 25 Monoids and groups Normal subgroups Algebra Interactive Theorem Normality Let H be a subgroup of G. The following asser- tions are equivalent. 1 g·H=H·g for every g∈G. −1 2 g·h·g ∈H for every g, h∈G. If H satisfies these properties, it is called a normal subgroup of G. A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 3 / 25 Monoids and groups Normal subgroups Algebra Interactive Theorem Normality Let H be a subgroup of G. The following asser- tions are equivalent. 1 g·H=H·g for every g∈G. −1 2 g·h·g ∈H for every g, h∈G. If H satisfies these properties, it is called a normal subgroup of G. A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 3 / 25 Monoids and groups Normal subgroups Algebra Example Interactive Symmetric groups: The alternating group Altn is a normal subgroup of Symn: If h is even, then −1 g·h·g is an even element of Symn for each g. Linear groups: SL(n, R) is a normal subgroup of GL(n, R): If det(A)=1, then for every invertible matrix B, the product B·A·B−1 has determinant 1. The center of a group: The center of a group is a normal subgroup since all its elements commute with every element in the group. Commutative groups: Suppose that G is a commutative group and H is a subgroup. Then for every g∈G and h∈H, we have g·h·g −1=h, so H is a normal subgroup of G. This shows that every subgroup of a commutative group is normal. A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 4 / 25 Monoids and groups Normal subgroups Algebra Example Interactive Symmetric groups: The alternating group Altn is a normal subgroup of Symn: If h is even, then −1 g·h·g is an even element of Symn for each g. Linear groups: SL(n, R) is a normal subgroup of GL(n, R): If det(A)=1, then for every invertible matrix B, the product B·A·B−1 has determinant 1. The center of a group: The center of a group is a normal subgroup since all its elements commute with every element in the group. Commutative groups: Suppose that G is a commutative group and H is a subgroup. Then for every g∈G and h∈H, we have g·h·g −1=h, so H is a normal subgroup of G. This shows that every subgroup of a commutative group is normal. A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 4 / 25 Monoids and groups Normal subgroups Algebra Example Interactive Symmetric groups: The alternating group Altn is a normal subgroup of Symn: If h is even, then −1 g·h·g is an even element of Symn for each g. Linear groups: SL(n, R) is a normal subgroup of GL(n, R): If det(A)=1, then for every invertible matrix B, the product B·A·B−1 has determinant 1. The center of a group: The center of a group is a normal subgroup since all its elements commute with every element in the group. Commutative groups: Suppose that G is a commutative group and H is a subgroup. Then for every g∈G and h∈H, we have g·h·g −1=h, so H is a normal subgroup of G. This shows that every subgroup of a commutative group is normal. A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 4 / 25 Monoids and groups Normal subgroups Algebra Example Interactive Symmetric groups: The alternating group Altn is a normal subgroup of Symn: If h is even, then −1 g·h·g is an even element of Symn for each g. Linear groups: SL(n, R) is a normal subgroup of GL(n, R): If det(A)=1, then for every invertible matrix B, the product B·A·B−1 has determinant 1. The center of a group: The center of a group is a normal subgroup since all its elements commute with every element in the group. Commutative groups: Suppose that G is a commutative group and H is a subgroup. Then for every g∈G and h∈H, we have g·h·g −1=h, so H is a normal subgroup of G. This shows that every subgroup of a commutative group is normal. A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 4 / 25 Monoids and groups Normal subgroups Algebra Example Interactive Symmetric groups: The alternating group Altn is a normal subgroup of Symn: If h is even, then −1 g·h·g is an even element of Symn for each g. Linear groups: SL(n, R) is a normal subgroup of GL(n, R): If det(A)=1, then for every invertible matrix B, the product B·A·B−1 has determinant 1. The center of a group: The center of a group is a normal subgroup since all its elements commute with every element in the group. Commutative groups: Suppose that G is a commutative group and H is a subgroup. Then for every g∈G and h∈H, we have g·h·g −1=h, so H is a normal subgroup of G. This shows that every subgroup of a commutative group is normal. A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 4 / 25 Monoids and groups Normal subgroups Algebra Interactive We will introduce computations modulo a normal subgroup and the corresponding construction of the quotient group. Let G be a group and let N be a normal subgroup of G. The notions of left coset [Cosets] (a set of the form g·N) and right coset [Cosets] (a set of the form N·g) of N in G coincide since normal subgroups satisfy g·N=N·g for all g∈G. Thus, we can just speak of cosets. A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 5 / 25 Monoids and groups Normal subgroups Algebra Interactive Theorem Suppose that N is a normal subgroup of G. Then, for all a, b∈G we have a·N·b·N=a·b·N; A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 6 / 25 Monoids and groups Normal subgroups Algebra Interactive Example Let G be the symmetric group Sym(3). The sub- group H=h(1, 2, 3)(1, 3, 2)iG of order 3 is a nor- mal subgroup. It has index 2. More generally, whenever H is a subgroup of G of index 2, it is a normal subgroup. For then, for g∈G, either g∈H and so g·H = H = H·g or or not, in which case g·H = G\H = H·g. A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 7 / 25 Monoids and groups Normal subgroups Algebra Interactive Remark Normal subgroups play the same role for groups as ideals do for rings. The procedure for making a quotient group is similar to the construction of a residue class ring. A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 8 / 25 Monoids and groups Normal subgroups Algebra Interactive Example Let G be the group of all motions in the plane. The subgroup T of all translations of the plane is a normal subgroup. Fix a point p of the plane. The subgroup H of G of all elements fixing the point p is a complement of T in the sense that H∩T ={1}. G=H·T . As a consequence, setwise G can be identified with the Cartesian product of H and T . But groupwise, it is not the direct product of these two groups. A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 9 / 25 Monoids and groups Normal subgroups Algebra Interactive Example Let G be the group of all motions in the plane. The subgroup T of all translations of the plane is a normal subgroup. Fix a point p of the plane. The subgroup H of G of all elements fixing the point p is a complement of T in the sense that H∩T ={1}. G=H·T . As a consequence, setwise G can be identified with the Cartesian product of H and T . But groupwise, it is not the direct product of these two groups. A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 9 / 25 Monoids and groups Normal subgroups Algebra Interactive Example Let G be the group of all motions in the plane.
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