Page 71 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
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mož Moravec: Some Topics in the Theory of Finite Groups 59
right) cosets of H in G is the index of H in G and is denoted by |G : H |. If G is a finite
group then Lagrange’s theorem says that |G | = |H | · |G : H |. In particular, if H ≤ G , then |H |
divides the order of G .
The intersection of a family of subgroups of a given group G is again a subgroup of
G . Thus, if X is a non-empty subset of G , then there exists the smallest subgroup of G
containing X . It is denoted by 〈X 〉 and called the subgroup generated by X . We say that a
group G is finitely generated if there exists a finite set X of its elements such that G = 〈X 〉.
Let G1 and G2 be groups. A map φ : G1 → G2 is said to be a homomorphism of groups
if it preserves group operation, that is,
(a b )φ = a φb φ for all a ,b ∈ G1,
where the products are calculated in the corresponding groups. The set
ker φ = {x ∈ G1 | x φ = 1}
is said to be the kernel of φ and is a subgroup of G1. The set
im φ = {x φ | x ∈ G1}
is a subgroup of G2 and is called the image of φ. A group homomorphism φ : G1 → G2 is
said to be an epimorphism if im φ = G2; monomorphism if ker φ = {1}; isomorphism if
it is epimorhism and monomorphism; endomorphism if G1 = G2. A bijective endomor-
phism is also called an automorphism.
A subgroup H of G is said to be a normal subgroup of G if x H = Hx for every x ∈ G .
Equivalently, x −1Hx ⊆ H for all x ∈ G , i.e., H is closed under conjugation by the elements
of G . If H is a normal subgroup of G then the sets of left and right cosets of H in G
coincide, and we use the commonly accepted notation G /H for these. The operation
on G /H given by Ha · Hb = H (a b ) is well defined and turns G /H into a group called
the factor group of G over H . The map ρ : G → G /H given by g ρ = H g is a surjective
homomorphism of groups with ker ρ = H .
The intersection of a family of normal subgroups of G is again a normal subgroup of
G . Thus, given a set X ⊆ G , there exists the smallest normal subgroup of G containing X .
It is denoted by 〈〈X 〉〉 and called the normal closure of X in G .
Theorem 3.2.1 (First Isomorphism Theorem) Let φ : G1 → G2 be a homomorphism of
groups. Then G1/ ker φ ∼= im φ.
Theorem 3.2.2 (Second Isomorphism Theorem) Let H be a subgroup and N a normal
subgroup of G . Then H ∩ N H , and H N /N =∼ H /(H ∩ N ).
Theorem 3.2.3 (Third Isomorphism Theorem) Let M and N be normal subgroups of G
and let N ≤ M . Then M /N G /N and (G /N )/(M /N ) ∼= G /M .
right) cosets of H in G is the index of H in G and is denoted by |G : H |. If G is a finite
group then Lagrange’s theorem says that |G | = |H | · |G : H |. In particular, if H ≤ G , then |H |
divides the order of G .
The intersection of a family of subgroups of a given group G is again a subgroup of
G . Thus, if X is a non-empty subset of G , then there exists the smallest subgroup of G
containing X . It is denoted by 〈X 〉 and called the subgroup generated by X . We say that a
group G is finitely generated if there exists a finite set X of its elements such that G = 〈X 〉.
Let G1 and G2 be groups. A map φ : G1 → G2 is said to be a homomorphism of groups
if it preserves group operation, that is,
(a b )φ = a φb φ for all a ,b ∈ G1,
where the products are calculated in the corresponding groups. The set
ker φ = {x ∈ G1 | x φ = 1}
is said to be the kernel of φ and is a subgroup of G1. The set
im φ = {x φ | x ∈ G1}
is a subgroup of G2 and is called the image of φ. A group homomorphism φ : G1 → G2 is
said to be an epimorphism if im φ = G2; monomorphism if ker φ = {1}; isomorphism if
it is epimorhism and monomorphism; endomorphism if G1 = G2. A bijective endomor-
phism is also called an automorphism.
A subgroup H of G is said to be a normal subgroup of G if x H = Hx for every x ∈ G .
Equivalently, x −1Hx ⊆ H for all x ∈ G , i.e., H is closed under conjugation by the elements
of G . If H is a normal subgroup of G then the sets of left and right cosets of H in G
coincide, and we use the commonly accepted notation G /H for these. The operation
on G /H given by Ha · Hb = H (a b ) is well defined and turns G /H into a group called
the factor group of G over H . The map ρ : G → G /H given by g ρ = H g is a surjective
homomorphism of groups with ker ρ = H .
The intersection of a family of normal subgroups of G is again a normal subgroup of
G . Thus, given a set X ⊆ G , there exists the smallest normal subgroup of G containing X .
It is denoted by 〈〈X 〉〉 and called the normal closure of X in G .
Theorem 3.2.1 (First Isomorphism Theorem) Let φ : G1 → G2 be a homomorphism of
groups. Then G1/ ker φ ∼= im φ.
Theorem 3.2.2 (Second Isomorphism Theorem) Let H be a subgroup and N a normal
subgroup of G . Then H ∩ N H , and H N /N =∼ H /(H ∩ N ).
Theorem 3.2.3 (Third Isomorphism Theorem) Let M and N be normal subgroups of G
and let N ≤ M . Then M /N G /N and (G /N )/(M /N ) ∼= G /M .
