Show that the multiplication table of a group is a latin square (each element appears exactly once in each row and column).
Answers
Step-by-step explanation:
The multiplication table of a finite group is a Latin square, when we label the entries. In other words, every row contains each element exactly once and every column contains each element exactly once. In fact, the multiplication table of a finite magma is a latin square if and only if that magma is a quasigroup.
In mathematics, the Latin square property is an elementary property of all groups and the defining property of quasigroups. It states that if (G, *) is a group or quasigroup and a and b are elements of G, then there exists a unique element x in G such that a*x=b, and a unique element y of G such that y*a=b.
Answer:
This is relatively easy to prove: Proposition. Any group table is a Latin square. Proof. Take a row indexed by the group element a.