28.6 SUBGROUPS
(a)IABCDE
I IABCDE
A AB IECD
B BIADEC
C CDE I AB
D DECB I A
E ECDAB I
(b)IABC
I IABC
A AICB
B BC I A
C CBAI
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) aboveholds is said to be amonomorphism(or an isomorphisminto), whilst a homomor-
phism for which (II) holds is called anepimorphism(or an isomorphismonto). 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 isomorphismG→G′is termed anautomorphism.
28.6 SubgroupsMore 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 asubgroup. A subgroupof a groupGcan be formally defined as any non-empty subsetH={Hi}of
G, the elements of which themselves behave as a group under the same rule of
combination as applies inGitself. As for all groups, the order of the subgroup is
equal to the number of elements it contains; we will denote it byhor|H|.
Any groupGcontains two trivial subgroups:(i)Gitself;
(ii) the setIconsisting of the identity element alone.All other subgroups ofGare termedproper subgroups. In a group with multipli-
cation 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-calledcyclicgroups, 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.