groups.py

def check_group(S, O, check_abelian=False):
    """Checks group properties on set S and operation O.

    Arguments:
        S (set): elements of this group
        O (callable): operation on two elements of S, returns an element of S
        check_abelian (bool): check if this is an abelian group

    """
    # check that S is closed under O
    for a in S:
        for b in S:
            x = O(a, b)
            y = O(b, a)
            assert x in S, ('error: S must be closed under O!', a, b, x)
            assert y in S, ('error: S must be closed under O!', a, b, y)

            if check_abelian:
                assert x == y, 'error: O(%s,%s) not commutative!' % (a, b)

    # check associativity
    for a in S:
        for b in S:
            for c in S:
                assert O(O(a, b), c) == O(a, O(b, c)), 'error: not associative!'

    # check identity element is unique
    is_id = lambda i, s: O(s, i) == O(i, s) == s
    identities = {i for i in S if all(is_id(i, b) for b in S)}
    assert len(identities) == 1, 'no identity element found!'
    e = identities.pop()

    # check inverse exists
    for a in S:
        invs_of_a = [b for b in S if O(a, b) == e and O(b, a) == e]
        assert len(invs_of_a) > 0, 'error: %s has no inverse!' % a

    return S, O


# trivial group
check_group({0}, lambda a, b: a)

# int_mod2 under addition
check_group({0, 1}, lambda a, b: (a + b) % 2)

# even_int_mod8 under addition
check_group({0, 2, 4, 6}, lambda a, b: (a + b) % 8)

# int_mod8 under addition
check_group({0, 1, 2, 3, 4, 5, 6, 7}, lambda a, b: (a + b) % 8)

# int_mod7 sans 0 under multiplication
check_group({1, 2, 3, 4, 5, 6}, lambda a, b: (a * b) % 7)