Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

52

ABSTRACT GROUPS

[CH. Ill

22. Definition of an Abstract Group and a Few Properties
of its Elements. In the theory of finite abstract groups we
deal with a set of distinct symbols G-s1, s2,..., so, and we
assume that any two of them can be combined according to
some law which is called multiplication, and which is denoted
in the same way as multiplication is commonly denoted. This
set of symbols represents a group provided the symbols satisfy
the following conditions:
1. If any two of the three symbols in an equation of the form
SSa = S-y
are contained in G, then the third is also contained in G, and it
is completely determined by this equation. It is assumed that
this statement includes the case when the two symbols which
are contained in G are identically equal to each other.
2. The symbols of G obey the associative law. That is,
(SaSO) Sy = Sa(SOSy).
From the former of these conditions it results that if one
of the three symbols in
SSOS = Sy
remains fixed while another assumes successively all the values
of the symbols or elements of G, the third will also run through
all these elements. In particular, there must be an element
sl such that
S1SB = Si.
Such an element is called the left-hand identity of so. This
left-hand identity is the same for all the elements of G. To see
this fact we multiply the last equation on the right by s, and
then let s, run through all the elements of G. In exactly the
same way it may be observed that G contains only one righthand identity s'.
To prove that s =sl it may be observed that if we replace
so by s' in the last equation it results that
s1s1= S1.