Page 72 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 72
3.2 Basic notions and examples
One can generalize the notion of normal subgroups as follows. A subgroup H of G
is said to be subnormal in G if there exists a finite series H = H0 H1 H2 · · · Hd = G .
The shortest length of such a series is called the defect of H in G . Subnormal subgroups
of defect one are precisely normal subgroups.
Two other notions related to normal subgroups are the following. A subgroup H of
G is said to be fully invariant if H α ≤ H for every endomorphism α of G . Similarly, H is
characteristic in G if H α ≤ H for every automorphism α of G . The following is straight-
forward:
Lemma 3.2.4 The properties of being a ‘characteristic subgroup’ and ‘fully invariant sub-
group’ are transitive relations. If H is characteristic in K and K normal in G then H G .
Let G be a group and x , y ∈ G . The commutator of x and y is defined by [x , y ] =
x −1y −1x y = x −1x y . The subgroup G generated by all the commutators [x , y ], where
x , y ∈ G , is called the derived subgroup or the commutator subgroup of G . Since [x , y ]α =
[x α, y α] for all endomorphisms α of G , it follows that G is a fully invariant subgroup of
G . It is easy to verify that G /G is abelian. Furthermore, if N is normal subgroup of G
with G /N abelian, then G ≤ N . Thus G /G can be seen as the largest abelian quotient of
G . It is called the abelianization of G . If G = G , then G is said to be a perfect group.
For a group G we define its center to be Z (G ) = {g ∈ G | [g , x ] = 1 for all x ∈ G }. It is
easy to verify that Z (G ) is a characteristic subgroup of G .
Let G1 and G2 be groups. The direct product G1 ×G2 is the group whose elements are
all pairs (g 1, g 2) ∈ G1 × G2, and the operation is given by
(a 1, a 2)(b1,b2) = (a 1b1, a 2b2).
Proposition 3.2.5 Let G , G1 and G2 be groups. Then G ∼= G1 × G2 if and only if there
exist normal subgroups H1 and H2 of G such that Hi ∼= Gi for i = 1, 2, H1 ∩ H2 = 1 and
H1H2 =G .
More generally, G ∼= G1×G2×· · ·×Gn if and only if there exist normal subgroups H1, . . . , Hn
of G such that Hi ∼= Gi , G = H1H2 · · · Hn , and
Hi ∩ H1 · · · Hi −1Hi +1 · · · Hn = {1}
for all i . This follows from Proposition 3.2.5 by induction.
Let X be a non-empty set, F a group, and ι : X → F a function. Then F , together
with ι, is said to be a free group on X if for each function α from X to a group G there
exists a homomorphism β : F → G such that α = ιβ . It is easy to show that ι has to be
injective. Up to isomorphism, there is precisely one free group on a given set X . It can be
constructed as a group whose elements are reduced words in X ∪ X −1, and the operation
One can generalize the notion of normal subgroups as follows. A subgroup H of G
is said to be subnormal in G if there exists a finite series H = H0 H1 H2 · · · Hd = G .
The shortest length of such a series is called the defect of H in G . Subnormal subgroups
of defect one are precisely normal subgroups.
Two other notions related to normal subgroups are the following. A subgroup H of
G is said to be fully invariant if H α ≤ H for every endomorphism α of G . Similarly, H is
characteristic in G if H α ≤ H for every automorphism α of G . The following is straight-
forward:
Lemma 3.2.4 The properties of being a ‘characteristic subgroup’ and ‘fully invariant sub-
group’ are transitive relations. If H is characteristic in K and K normal in G then H G .
Let G be a group and x , y ∈ G . The commutator of x and y is defined by [x , y ] =
x −1y −1x y = x −1x y . The subgroup G generated by all the commutators [x , y ], where
x , y ∈ G , is called the derived subgroup or the commutator subgroup of G . Since [x , y ]α =
[x α, y α] for all endomorphisms α of G , it follows that G is a fully invariant subgroup of
G . It is easy to verify that G /G is abelian. Furthermore, if N is normal subgroup of G
with G /N abelian, then G ≤ N . Thus G /G can be seen as the largest abelian quotient of
G . It is called the abelianization of G . If G = G , then G is said to be a perfect group.
For a group G we define its center to be Z (G ) = {g ∈ G | [g , x ] = 1 for all x ∈ G }. It is
easy to verify that Z (G ) is a characteristic subgroup of G .
Let G1 and G2 be groups. The direct product G1 ×G2 is the group whose elements are
all pairs (g 1, g 2) ∈ G1 × G2, and the operation is given by
(a 1, a 2)(b1,b2) = (a 1b1, a 2b2).
Proposition 3.2.5 Let G , G1 and G2 be groups. Then G ∼= G1 × G2 if and only if there
exist normal subgroups H1 and H2 of G such that Hi ∼= Gi for i = 1, 2, H1 ∩ H2 = 1 and
H1H2 =G .
More generally, G ∼= G1×G2×· · ·×Gn if and only if there exist normal subgroups H1, . . . , Hn
of G such that Hi ∼= Gi , G = H1H2 · · · Hn , and
Hi ∩ H1 · · · Hi −1Hi +1 · · · Hn = {1}
for all i . This follows from Proposition 3.2.5 by induction.
Let X be a non-empty set, F a group, and ι : X → F a function. Then F , together
with ι, is said to be a free group on X if for each function α from X to a group G there
exists a homomorphism β : F → G such that α = ιβ . It is easy to show that ι has to be
injective. Up to isomorphism, there is precisely one free group on a given set X . It can be
constructed as a group whose elements are reduced words in X ∪ X −1, and the operation
