3210.sagews

A = matrix([[0,1,2],[1,2,0],[2,0,1]])
def semi(n,A):
    c = True
    for i in range(n):
        for j in range(n):
            for k in range(n):
                a = A[i, j]
                b = A[j, k]
                c = (A[a, k] == A[i, b]) and c
    return c
def id(n,A):
    c = True
    d = False
    for i in range(n):
        for j in range(n):
            k = (i+j) % n
            c = (k == A[k, i]) and (k == A[i, k]) and c
        d = c or d
    return d
def group(n,id,A):
    c = False
    d = True
    for i in range(n):
        for j in range(n):
            c = ((id == A[j, i]) and (id == A[i, j])) or c
        d = c and d
    return d
def abel(n,A):
    d = True
    for i in range(n):
        for j in xrange(i,n):
            c = A[j, i] == A[i, j]
            d = c and d
    return d
print semi(3,A)
print id(3,A)
print group(3,0,A)
print abel(3,A)

True True True True