Automatically find all homomorphisms between (Z/nZ, *) and (Z/mZ, *). This can easily be adjusted to find homomorphisms between (Z/nZ, +) and (Z/mZ, +)

findHomomorphisms.py

#!/usr/bin/env python
# -*- coding: utf-8 -*-

def isGroupHomomorphism(phi, group1, group2, verbose=False):
	for a, b in product(group1['set'], group2['set']):
		inner = group1['operator'](a, b)
		outer = group2['operator'](phi(a), phi(b))
		if verbose:
			print("f(%i) = %i, f(%i) = %i       f(%i)=%i=%i" % (a, phi(a), b, phi(b), inner, phi(inner), outer))
		if phi(inner) != outer:
			return False
	return True

def getZModPlusGroup(n):
	group = {}
	group['name'] = "Z/" + str(n) + "Z"
	group['set'] = list(range(1,n))
	group['operator'] = lambda x, y: (x*y)%n
	return group

if __name__ == "__main__":
	group1 = getZModPlusGroup(5)
	group2 = getZModPlusGroup(5)

	from itertools import product
	checkedCounter = 0
	homCounter = 0
	candidates = len(group2['set'])**len(group1['set'])
	if candidates > 10000000:
		print("Warning! This will test %i candidates!" % candidates)
	for t in product(group2['set'], repeat=len(group1['set'])): #mapping from group1 to elements from group2
		checkedCounter += 1
		phi = lambda x: t[x-1] #* / +
		if 1==2 and phi(0) != 0:
			continue
		else:
			if isGroupHomomorphism(phi, group1, group2):
				#isGroupHomomorphism(phi, group1, group2, True)
				homCounter += 1
				print(group1['name'] + ":\t" + str(group1['set']))
				print(group2['name'] + ":\t" + str(t))
				print "#"*25
	print("Checked %i candidates. %i of them were homomorphisms." % (checkedCounter, homCounter))