COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

232 Takayuki Tamura
8” = 0B if and only if c&rE~(8). Let ?$ = {G; acQ(G)}. Then G, is
isomorphic onto G(G) under a - a. Suppose a = ,9. x8-l is in %(G, 0)
for all BE@. On the other hand, there is 80~@ such that $?l(G, f3,) consists
of the identical mapping E alone (cf. [ZO]). Hence zB-1 = e and so K = b.
Define another unary operation 0 + 8 on @ as follows:
xe’y = yex.
Then clearly (0’)’ = 0 and (G, 0) is anti-isomorphic onto (G, 0’); 8’ = 8
if and only if (G, 0) is commutative. Also (e’y = (fP)’ for all 0 c 8.
We denote (e’y by 8”‘. Then a is an anti-automorphism of (G, 0)
if and only if 8”’ = 8. We can easily prove
(ey = 03
(e*y = fjcv
(eqg = eQ.
As defined in 9 1 .I, 58(e) is the automorphism group of (G, 0) while g’(0)
denotes the set of all anti-automorphisms of (G, 0).
We define
we) = ate) u we).
Then %(e) is a subgroup of E(G, 0) and the index of a(0) to B(8) is 2.
Let BEG(G). Then
ate4 = ~--2w~~p, myeq = pa’(e)q3.
Let @ = %(G, 0) and let aEG(G). Then @ = a(P) if and only if a is
in the normalizer ‘8(Q) of .Q in E?(G). Therefore 8” = @ and a(@) =
%(@) = a(e) = @ if and only if a, ,!I E g(Q) and tc F p (mod Q).
Let $I be a permutation group over a set G and suppose that Q is generated
by a subset 8 = {ad; 1. E X} of 8.
Let
GXG = {(KY); x,ycG}.
A binary operation on G is understood to be a mapping 0 of GX G into G.
$ is contained in the automorphism group ‘8(G) of a groupoid G defined
by 0 if and only if, for X, y E G,
[(x, y)e]K = (xa, ycr)e for all a($.
We define an equivalence relation % on GX G as follows:
(x, y) 23 (z, u) if and only if z = xa, u = ya for some tc E 8. Clearly %
is the transitive closure of a relation %r, defined by
(x, y) ‘@I (z, u) if and only if z = xa, u = ya for some a E 9.
If we let c = (a, b)B and if (x, y) %(a, b), then (x, y)S is automatically
determined by
(x, y)e = [(a, 6)0]a for some a EQ.
Let {(a,, b,); 5 E E} be a representative system from the equivalence classes
modulo %. We may determine only {(a(, b,)B; 5 ~8). However, there
Semigroups and groupoids 233
is some restriction for choosing (at, b$:
[(at, &)B]a = (acGc, bta)e.
For (at, be) define an equivalence relation nJ on the set union e”i?-’
5
as follows :
“;/!I i f an donly
if (a,-a, b(a) = (atb, b&).
For (aE, b,) we select an element c: o f G such that the following condition
is satisfied :
a ; fl implies cga = c$.
1.4. Groupoids of order == 3. First of all we explain the notation and the
abbreviations appearing below :
sj
si
c
II
up to is0
Automorphism group @.
The symmetric group of degree i.
The number of conjugates of !$,
I &I
i’e’ ’ = 1 Normalizer 1 *
The index of Q to its normalizer,
1 Normalizer ]
n= -
lsjl *
Up to isomorphism.
up to dual Up to dual-isomorphism (i.e. anti-isomorphism).
self-dual
comm
Anti-isomorphic to itself.
Commutative.
First we have the following table for groupoids of order 2. Since the
case is simple, we omit the explanation.
Total
TABLE 1. Groupoids of Order 2
1 I I