Does the following binary operation form a group on a set with 3 elements? (multiple identities?)

Question

Let S = {a, b, c}.

*| a  b  c
-----------
a| a  b  c
b| b  a  a
c| c  a  a

This seems to have all the desired characteristics of a group, however, both b and c have two identities. Is (S, *) a group?

Thanks!

Answer:So if one takes element a∈S to be the identity there is actually only one operation that can be can be defined to make (S, *) a group, as each element can only appear in each column and row once. Namely:

*| a  b  c
-----------
a| a  b  c
b| b  c  a
c| c  a  b

Answer 1

No, since there's a row with repeated elements. In particular, the given operation is not associative, since: $$ b * (c * c) = b * a = b \neq c = a * c = (b * c) * c $$

Answer 2

In every group $(G,\cdot)$, for all $p,q\in G$ the equation $x \cdot p = q$ has a unique solution for $x$, and similarly for $p \cdot x = q$ (although it need not be the same $x$). This means that the group table must be a "magic square", i.e. each element must occur precisely once in each row and each column.