Subgroups

28.6 SUBGROUPS
(a)
I
A
B
C
D
E
I
I
A
B
C
D
E
A
A
B
I
D
E
C
B
B
I
A
E
C
D
C
C
E
D
I
B
A
D
D
C
E
A
I
B
E
E
D
C
B
A
I
(b)
I
A
B
C
I
I
A
B
C
A
A
I
C
B
B
B
C
I
A
C
C
B
A
I
Table 28.9 Reproduction of (a) table 28.8 and (b) table 28.3 with the relevant
subgroups shown in bold.
For the sake of completeness, we add that a homomorphism for which (I) above
holds is said to be a monomorphism (or an isomorphism into), whilst a homomorphism for which (II) holds is called an epimorphism (or an isomorphism onto). If,
in either case, the other requirement is met as well then the monomorphism or
epimorphism is also an isomorphism.
Finally, if the initial and final groups are the same, G = G , then the isomorphism
G → G is termed an automorphism.
28.6 Subgroups
More detailed inspection of tables 28.8 and 28.3 shows that not only do the
complete tables have the properties associated with a group multiplication table
(see section 28.2) but so do the upper left corners of each table taken on their
own. The relevant parts are shown in bold in the tables 28.9(a) and (b).
This observation immediately prompts the notion of a subgroup. A subgroup
of a group G can be formally defined as any non-empty subset H = {Hi } of
G, the elements of which themselves behave as a group under the same rule of
combination as applies in G itself. As for all groups, the order of the subgroup is
equal to the number of elements it contains; we will denote it by h or |H|.
Any group G contains two trivial subgroups:
(i) G itself;
(ii) the set I consisting of the identity element alone.
All other subgroups of G are termed proper subgroups. In a group with multiplication table 28.8 the elements {I, A, B} form a proper subgroup, as do {I, A} in a
group with table 28.3 as its group table.
Some groups have no proper subgroups. For example, the so-called cyclic
groups, mentioned at the end of subsection 28.1.1, have no subgroups other
than the whole group or the identity alone. Tables 28.10(a) and (b) show the
multiplication tables for two of these groups. Table 28.6 is also the group table
for a cyclic group, that of order 4.
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GROUP THEORY
(a)
I
A
B
I
I
A
B
A
A
B
I
B
B
I
A
(b)
I
A
B
C
D
I
I
A
B
C
D
A
A
B
C
D
I
B
B
C
D
I
A
C
C
D
I
A
B
D
D
I
A
B
C
Table 28.10 The group tables of two cyclic groups, of orders 3 and 5. They
have no proper subgroups.
It will be clear that for a cyclic group G repeated combination of any element
with itself generates all other elements of G, before finally reproducing itself. So,
for example, in table 28.10(b), starting with (say) D, repeated combination with
itself produces, in turn, C, B, A, I and finally D again. As noted earlier, in any
cyclic group G every element, apart from the identity, is of order g, the order of
the group itself.
The two tables shown are for groups of orders 3 and 5. It will be proved in
subsection 28.7.2 that the order of any group is a multiple of the order of any of
its subgroups (Lagrange’s theorem), i.e. in our general notation, g is a multiple
of h. It thus follows that a group of order p, where p is any prime, must be cyclic
and cannot have any proper subgroups. The groups for which tables 28.10(a) and
(b) are the group tables are two such examples. Groups of non-prime order may
(table 28.3) or may not (table 28.6) have proper subgroups.
As we have seen, repeated multiplication of an element X (not the identity)
by itself will generate a subgroup {X, X 2 , X 3 , . . . }. The subgroup will clearly be
Abelian, and if X is of order m, i.e. X m = I, the subgroup will have m distinct
members. If m is less than g – though, in view of Lagrange’s theorem, m must
be a factor of g – the subgroup will be a proper subgroup. We can deduce, in
passing, that the order of any element of a group is an exact divisor of the order
of the group.
Some obvious properties of the subgroups of a group G, which can be listed
without formal proof, are as follows.
(i) The identity element of G belongs to every subgroup H.
(ii) If element X belongs to a subgroup H, so does X −1 .
(iii) The set of elements in G that belong to every subgroup of G themselves
form a subgroup, though this may consist of the identity alone.
Properties of subgroups that need more explicit proof are given in the following sections, though some need the development of new concepts before they
can be established. However, we can begin with a theorem, applicable to all
homomorphisms, not just isomorphisms, that requires no new concepts.
Let Φ : G → G be a homomorphism of G into G ; then
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