Groups
Our readers will have met groups before, so we shall be fairly brief in recalling the
fundamentals, which occupy Sections 2.1-2.3. The remainder of this chapter deals
with some notions of importance in elucidating the structure of groups, such as
solubility, nilpotence (Section 2.4) and commutator subgroups (Section 2.5). In
Section 2.6 we describe the constructions of Frattini and Fitting, which have their
counterpart in rings in the form of the radical.
2.1 Definition and Basic Properties
We recall that a group is a set G on which a binary operation is defined, with values in
G, denoted by x.y or simply xy, the product, such that
G.l (xy)z = x(yz) for all x, y, z E G (associative law).
G.2 There exists an element e E G such that xe = ex = x for all x E G.
G.3 For each x E G there exists Xl E G such that XXi = XiX = e.
The element e in G.2 is uniquely determined by the equations in G.2; it is called the
neutral element or unit element, and is usually denoted by e or 1. The element Xl in
G.3 is uniquely determined by x; it is called the inverse of x and is denoted by x - I.
If the commutative law holds in G: xy = yx for all x, y E G, the group is said to be
abelian. For an abelian group the additive notation is sometimes used for the opera-
tion; it is then called the sum and denoted by x + y, the neutral element is written 0
(and called the zero element), and the inverse of x is written -x.
As a consequence of G.l, the value of a repeated product XIX2 ... Xn is independent
of bracketing, as long as the order of the factors is preserved. This follows easily by
induction on n: for n = 3 it is just G.l; when n > 3, suppose that the product is
bracketed in two different ways. We must show that
(2.1.1)
and we may suppose that 1 < i < j :::; n. Moreover the portions inside the brackets
are independent of the bracketing, by the induction hypothesis. Writing
u = XI' .. Xi-I> V = Xi ... Xj_1> W = Xj . .. Xm we can rewrite (2.1.1) as u( vw) = (uv)w,
and this is seen to follow from G.l. Thus XIX2 ... Xn is well-defined. In particular,
25
,26 Groups
taking all the Xi equal to X, we obtain xx . .. X (n factors), which is denoted by xn. This
can be defined even for negative n by putting X - n = (x - I)" . We note the rules for
any group element x and any m, n E Z:
(2.1.2)
whose verification is straightforward. Similarly, in additive notation, x + x + ... + x
with n terms is written as nx. A group G is called elementary abelian if for some prime
p, px = 0 for all x E G. An example is the abelian group consisting of four elements
e, a, b, c with a2 = b2 = c2 = e, ab = c; it is known as the Klein 4-group.
Probably the most familiar example is the additive group of integers, with neutral
element 0 and -x as the inverse of x; another example is the multiplicative group of
non-zero rational numbers with neutral element 1 and X-I as the inverse of x. An
even simpler example is the trivial group, consisting of a single element e with multi-
plication ee = e; the trivial group will usually be denoted by 1.
The groups mentioned so far are all abelian; to obtain a non-abelian group, recall
that for any set S, by a permutation of S one understands a bijective mapping from S
to itself. It is clear that for any set S the set ~(S) of all permutations of S is a group
under composition of mappings, known as the symmetric group on S; the cardinal of
S is called the degree of ~(S). It is not hard to see that any symmetric group of degree
greater than two is non-abelian.
Let G be a group; a subgroup of G is a subset H of G which is a group relative to the
operations of G. Thus H is a subgroup iff 1 E H and for any x, y E H, xy, X -I E H.
An element c satisfying cn = 1 for some n ::: 1 is said to be of finite order, and the
least such n is called the order of c. Clearly in a finite group every element is of
finite order, but there are also infinite groups with this property; they are usually
called torsion groups. In an abelian group the set of all elements of finite order is a
subgroup, called the torsion subgroup.
It is easily verified that the intersection of two subgroups is again a subgroup.
More generally, this holds for any set of subgroups. Given a subset X of a group,
the set consisting of 1 and all finite products of members of X and their inverses,
briefly group words in X, forms a subgroup, denoted by gp{X} and called the sub-
group generated by X. Clearly any subgroup containing X also contains gp{X};
it follows that gp{X} is equal to the intersection of all subgroups containing X.
If G is a group generated by a set X, then any equation of the form f = g, where
f, g are group words in X, is called a relation. Let G be generated by X and let R
be a set of relations such that every relation in the elements of X holding in G is a
consequence of relations in R; then we write G = gp{XIR} and call this a presentation
of G by X as generating set with R as set of defining relations. In a relation of the form
f = 1, f is also called a relator. For example, the quaternions i, j generate a group of
order 8, the quaternion group, which has the presentation gp{a, bla4 = e, a2 = b2 ,
b - lab = a -I }. The group of motions of a regular n-gon, which consists of n rota-
tions and n reflexions, has the presentation gp{a, bla n = b2 = e, b-Iab = a-I}; it is
called the dihedral group of order 2n and is denoted by Dn.
Let G, H be any groups. A mapping f : G -+ H is called a homomorphism if
(xy)f = xf. yf for all x, y E G; it follows that IGf = IH, where I G , IH denote the
unit elements of G, H respectively. The set mapped to 1 : {x E Glxf = I} is a sub-
, 2.1 Definition and Basic Properties 27
group called the kernel of f When H = G, we call f an endomorphism of G. If
f : G -+ H is bijective, it is called an isomorphism and G is said to be isomorphic
to H; clearly f- I is then an isomorphism from H to G. An isomorphism of G
with itself is called an automorphism. For example, in any group G, the mapping
x 1-+ c - I xc( c E G) is an automorphism, the inner automorphism or conjugation by c.
A group generated by a single element is said to be cyclic. For example, the integers
Z form a cyclic group under addition, with 1 as generator. Any cyclic group is a
homomorphic image of Z; the kernel is a subgroup of Z, and hence of the form
mZ, for some m E Z, m ::: O. Thus every cyclic group is either isomorphic to Z or
of the form Zjm = {xlmx = O} (m EN). In multiplicative notation this is the
group of m-th roots of 1; so up to isomorphism there is one infinite cyclic group
and one cyclic group of order m, for each positive integer m. It will usually be
denoted by em or Z/m.
For any group G, the subgroups of G that are maximal among the proper sub-
groups of G are simply called the maximal subgroups. For finitely generated (non-
trivial) groups their existence is guaranteed by Zorn's lemma:
Proposition 2.1.1. Let G be a finitely generated non-trivial group and H be a proper
subgroup. Then there is a maximal subgroup containing H.
Proof. Let X be a finite generating set of G and consider the set f!J> of all proper sub-
groups containing H. Given any chain in f!J>, its union A is clearly a subgroup of G; if
A = G, then A ;2 X and since X is finite, all the members of X belong to some
member B of this chain; but this would mean that B = G, a contradiction. This
shows that A must be proper, hence f!J> is inductive and by Zorn's lemma it has a
maximal member, which is the desired subgroup. •
Given groups G, H, if f : G -+ H is a general homomorphism, the image Gf of G
is easily seen to be a subgroup of H, the homomorphic image of Gunder f. The inverse
image If-I of the unit element of H, the kernel of f, written ker f, is a normal sub-
group of G, i.e. it is a subgroup such that x - I (ker f )x = ker f for all x E G. This
follows because if af = 1, then (x-Iax)f = (x!) -1 1(x!) = 1 for all x E G.
A group G with two subgroups H, K is said to be the direct product of H and K
if every element g E G can be uniquely expressed as g = xy = yx, where x E H,
Y E K. It follows that H, K are both normal in G, HK = G and H n K = 1.
Conversely, any two subgroups H, K of G satisfying these conditions give rise to a
direct product representation of G.
Let G be any group and H be a subgroup; any set of the form Ha = {halh E H} is
called a right coset of H in G. If two cosets Ha, Hb have a common element ua = vb
(where u, v E H), say, then for any hE H, ha = hu-Ivb, hb = hv-Iua and it follows
that Ha = Hb. Thus the right cosets of H in G form a partition of G. A left coset of H
is similarly defined as aH = {ahlh E H}. We shall denote the set ofright cosets of H
in G by G/H and the set of left cosets by mG. In general left cosets define a different
partition of G, but their number is the same, since the correspondence
(2.1.3)
Our readers will have met groups before, so we shall be fairly brief in recalling the
fundamentals, which occupy Sections 2.1-2.3. The remainder of this chapter deals
with some notions of importance in elucidating the structure of groups, such as
solubility, nilpotence (Section 2.4) and commutator subgroups (Section 2.5). In
Section 2.6 we describe the constructions of Frattini and Fitting, which have their
counterpart in rings in the form of the radical.
2.1 Definition and Basic Properties
We recall that a group is a set G on which a binary operation is defined, with values in
G, denoted by x.y or simply xy, the product, such that
G.l (xy)z = x(yz) for all x, y, z E G (associative law).
G.2 There exists an element e E G such that xe = ex = x for all x E G.
G.3 For each x E G there exists Xl E G such that XXi = XiX = e.
The element e in G.2 is uniquely determined by the equations in G.2; it is called the
neutral element or unit element, and is usually denoted by e or 1. The element Xl in
G.3 is uniquely determined by x; it is called the inverse of x and is denoted by x - I.
If the commutative law holds in G: xy = yx for all x, y E G, the group is said to be
abelian. For an abelian group the additive notation is sometimes used for the opera-
tion; it is then called the sum and denoted by x + y, the neutral element is written 0
(and called the zero element), and the inverse of x is written -x.
As a consequence of G.l, the value of a repeated product XIX2 ... Xn is independent
of bracketing, as long as the order of the factors is preserved. This follows easily by
induction on n: for n = 3 it is just G.l; when n > 3, suppose that the product is
bracketed in two different ways. We must show that
(2.1.1)
and we may suppose that 1 < i < j :::; n. Moreover the portions inside the brackets
are independent of the bracketing, by the induction hypothesis. Writing
u = XI' .. Xi-I> V = Xi ... Xj_1> W = Xj . .. Xm we can rewrite (2.1.1) as u( vw) = (uv)w,
and this is seen to follow from G.l. Thus XIX2 ... Xn is well-defined. In particular,
25
,26 Groups
taking all the Xi equal to X, we obtain xx . .. X (n factors), which is denoted by xn. This
can be defined even for negative n by putting X - n = (x - I)" . We note the rules for
any group element x and any m, n E Z:
(2.1.2)
whose verification is straightforward. Similarly, in additive notation, x + x + ... + x
with n terms is written as nx. A group G is called elementary abelian if for some prime
p, px = 0 for all x E G. An example is the abelian group consisting of four elements
e, a, b, c with a2 = b2 = c2 = e, ab = c; it is known as the Klein 4-group.
Probably the most familiar example is the additive group of integers, with neutral
element 0 and -x as the inverse of x; another example is the multiplicative group of
non-zero rational numbers with neutral element 1 and X-I as the inverse of x. An
even simpler example is the trivial group, consisting of a single element e with multi-
plication ee = e; the trivial group will usually be denoted by 1.
The groups mentioned so far are all abelian; to obtain a non-abelian group, recall
that for any set S, by a permutation of S one understands a bijective mapping from S
to itself. It is clear that for any set S the set ~(S) of all permutations of S is a group
under composition of mappings, known as the symmetric group on S; the cardinal of
S is called the degree of ~(S). It is not hard to see that any symmetric group of degree
greater than two is non-abelian.
Let G be a group; a subgroup of G is a subset H of G which is a group relative to the
operations of G. Thus H is a subgroup iff 1 E H and for any x, y E H, xy, X -I E H.
An element c satisfying cn = 1 for some n ::: 1 is said to be of finite order, and the
least such n is called the order of c. Clearly in a finite group every element is of
finite order, but there are also infinite groups with this property; they are usually
called torsion groups. In an abelian group the set of all elements of finite order is a
subgroup, called the torsion subgroup.
It is easily verified that the intersection of two subgroups is again a subgroup.
More generally, this holds for any set of subgroups. Given a subset X of a group,
the set consisting of 1 and all finite products of members of X and their inverses,
briefly group words in X, forms a subgroup, denoted by gp{X} and called the sub-
group generated by X. Clearly any subgroup containing X also contains gp{X};
it follows that gp{X} is equal to the intersection of all subgroups containing X.
If G is a group generated by a set X, then any equation of the form f = g, where
f, g are group words in X, is called a relation. Let G be generated by X and let R
be a set of relations such that every relation in the elements of X holding in G is a
consequence of relations in R; then we write G = gp{XIR} and call this a presentation
of G by X as generating set with R as set of defining relations. In a relation of the form
f = 1, f is also called a relator. For example, the quaternions i, j generate a group of
order 8, the quaternion group, which has the presentation gp{a, bla4 = e, a2 = b2 ,
b - lab = a -I }. The group of motions of a regular n-gon, which consists of n rota-
tions and n reflexions, has the presentation gp{a, bla n = b2 = e, b-Iab = a-I}; it is
called the dihedral group of order 2n and is denoted by Dn.
Let G, H be any groups. A mapping f : G -+ H is called a homomorphism if
(xy)f = xf. yf for all x, y E G; it follows that IGf = IH, where I G , IH denote the
unit elements of G, H respectively. The set mapped to 1 : {x E Glxf = I} is a sub-
, 2.1 Definition and Basic Properties 27
group called the kernel of f When H = G, we call f an endomorphism of G. If
f : G -+ H is bijective, it is called an isomorphism and G is said to be isomorphic
to H; clearly f- I is then an isomorphism from H to G. An isomorphism of G
with itself is called an automorphism. For example, in any group G, the mapping
x 1-+ c - I xc( c E G) is an automorphism, the inner automorphism or conjugation by c.
A group generated by a single element is said to be cyclic. For example, the integers
Z form a cyclic group under addition, with 1 as generator. Any cyclic group is a
homomorphic image of Z; the kernel is a subgroup of Z, and hence of the form
mZ, for some m E Z, m ::: O. Thus every cyclic group is either isomorphic to Z or
of the form Zjm = {xlmx = O} (m EN). In multiplicative notation this is the
group of m-th roots of 1; so up to isomorphism there is one infinite cyclic group
and one cyclic group of order m, for each positive integer m. It will usually be
denoted by em or Z/m.
For any group G, the subgroups of G that are maximal among the proper sub-
groups of G are simply called the maximal subgroups. For finitely generated (non-
trivial) groups their existence is guaranteed by Zorn's lemma:
Proposition 2.1.1. Let G be a finitely generated non-trivial group and H be a proper
subgroup. Then there is a maximal subgroup containing H.
Proof. Let X be a finite generating set of G and consider the set f!J> of all proper sub-
groups containing H. Given any chain in f!J>, its union A is clearly a subgroup of G; if
A = G, then A ;2 X and since X is finite, all the members of X belong to some
member B of this chain; but this would mean that B = G, a contradiction. This
shows that A must be proper, hence f!J> is inductive and by Zorn's lemma it has a
maximal member, which is the desired subgroup. •
Given groups G, H, if f : G -+ H is a general homomorphism, the image Gf of G
is easily seen to be a subgroup of H, the homomorphic image of Gunder f. The inverse
image If-I of the unit element of H, the kernel of f, written ker f, is a normal sub-
group of G, i.e. it is a subgroup such that x - I (ker f )x = ker f for all x E G. This
follows because if af = 1, then (x-Iax)f = (x!) -1 1(x!) = 1 for all x E G.
A group G with two subgroups H, K is said to be the direct product of H and K
if every element g E G can be uniquely expressed as g = xy = yx, where x E H,
Y E K. It follows that H, K are both normal in G, HK = G and H n K = 1.
Conversely, any two subgroups H, K of G satisfying these conditions give rise to a
direct product representation of G.
Let G be any group and H be a subgroup; any set of the form Ha = {halh E H} is
called a right coset of H in G. If two cosets Ha, Hb have a common element ua = vb
(where u, v E H), say, then for any hE H, ha = hu-Ivb, hb = hv-Iua and it follows
that Ha = Hb. Thus the right cosets of H in G form a partition of G. A left coset of H
is similarly defined as aH = {ahlh E H}. We shall denote the set ofright cosets of H
in G by G/H and the set of left cosets by mG. In general left cosets define a different
partition of G, but their number is the same, since the correspondence
(2.1.3)
