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. 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
Due to these properties, the set G/N of cosets admits the following group structure.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 10 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Definition We call the group G/N with multiplication: g·N·g‘·N=g·g‘·N unit: N inverse: g·N→g −1·N the quotient group of G with respect to N.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 11 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Definition We call the group G/N with multiplication: g·N·g‘·N=g·g‘·N unit: N inverse: g·N→g −1·N the quotient group of G with respect to N.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 11 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Definition We call the group G/N with multiplication: g·N·g‘·N=g·g‘·N unit: N inverse: g·N→g −1·N the quotient group of G with respect to N.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 11 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Definition We call the group G/N with multiplication: g·N·g‘·N=g·g‘·N unit: N inverse: g·N→g −1·N the quotient group of G with respect to N.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 11 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Definition We call the group G/N with multiplication: g·N·g‘·N=g·g‘·N unit: N inverse: g·N→g −1·N the quotient group of G with respect to N.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 11 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Example The additive group of Q is commutative. There- fore, the subgroup Z is a normal subgroup of Q. The cosets of Z in Q are the sets of the form a b + Z, where a, b in Z and b6=0. For example, 1 2 + Z. Computing in the quotient Q/Z comes down to ‘computing modulo integers‘. For exam- 3 5 7 ple 4 + Z + 6 + Z= 12 + Z.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 12 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Example If G is a commutative group, then each subgroup H of G is a normal subgroup. Thus, the quotient group G/H always exists. Moreover, it is com- mutative.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 13 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Computing modulo a normal subgroup behaves well, as becomes clear by the following result.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 14 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Theorem Let N be a normal subgroup of the group G. The map p : G→G/N, g 7→ g·N is a surjective homomorphism with kernel N.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 15 / 25 Monoids and groups Normal subgroups
Algebra Interactive
In Theorem ?? it was shown that the kernel of a group homomorphism is a normal subgroup. Theorem 15 states the converse, namely that ev- ery normal subgroup is the kernel of a homomor- phism.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 16 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Example Let G be the set of all 2×2 matrices with entries 1 x from a field F of the form , where x is an 0 y arbitrary element of F and y a nonzero element of F . Then G is a subgroup of GL(2, F ). The 1 x subgroup N of all matrices of the form 0 1 is a normal subgroup of G. The quotient group G/N is isomorphic to the multiplicative group on F \{0}. Observe that N is the kernel of the determinant, viewed as a homomorphism.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 17 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Let f : G→H be a surjective group homomor- phism with kernel N. According to a previous proposition, N is a normal subgroup of G.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 18 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Theorem First isomorphism theorem for groups If G and H are groups and f : G→H is a surjec- tive homomorphism with kernel N, then the map f 0 : G/N→H defined by f 0(g·N)=f (g) is an iso- morphism.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 19 / 25 Monoids and groups Normal subgroups
Algebra Interactive
According to the theorem, computing in H is es- sentially the same as computing in G/N.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 20 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Example The classification of cyclic groups can be han- dled easily with the theorem. Because G is cyclic, there exists g∈G with hgi{g}=G. Consider the map f : Z→G, i 7→ g i . It is a surjective homo- morphism with kernel n·Z for some non-negative integer n. The assertion that every cyclic group is isomorphic to either Z (the case where n=0) or Cn for some positive integer n now follows directly from the First isomorphism theorem for groups 19 applied to f .
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 21 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Example The cyclic group of order n Since Z is commutative, its subgroup n·Z is nor- mal. It is the kernel of the surjective homomor- phism. The quotient of this group is the group Cn.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 22 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Example Different groups with same quotient and ker- nel Let G be a group and N a normal subgroup of G distinct from 1 and from G. The groups G/N and N are both smaller than G. A lot of information about G can be obtained from study of these two smaller groups. However, the exact structure of G is not completely determined by G/N and N. For instance, the groups C4 and C2 × C2 both have a normal subgroup isomorphic with C2, and in both cases the quotient group is isomorphic with C2.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 23 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Example The quotient of the symmetric group by the alternating group
The group Symn/Altn is isomorphic with C2. For, the map sgn: Symn→{1, −1} is a surjective ho- momorphism of groups with kernel Altn. Here, {1, −1} is the group of invertible elements of the monoid [Z, ·, 1]. This group is isomorphic with C2.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 24 / 25 Monoids and groups Normal subgroups
Algebra Interactive
Example The general linear group The quotient group GL(n, R)/SL(n, R) is iso- morphic to the multiplicative group R×. The subgroup SL(n, R) is the kernel of the determi- nant map det: GL(n, R)→R×.
A.M. Cohen, H. Cuypers, H. Sterk Algebra 2 September 25, 2006 25 / 25