'''
This library exists to isolate all math functions
related to groups and their representations. 
'''

from math import gcd

'''
Returns the multiplicative cyclic subgroup 
generated by an element g modulo m.
Returns the cyclic subgroup as a list[int],
    the order of that subgroup, and a boolean
    indicating whether g is infinitely repeating
    with period == ord<g> (or otherwise if it
    terminates with g**ord<g> == 0).
'''
def cyclic_subgrp(g: int, 
                  m: int, 
                  ignore_zero: bool = True) -> tuple[list[int], int, bool]:
    G = []
    order = 0
    periodic = True
    a = 1 # start at identity
    for _ in range(m):
        a = (a * g) % m
        if a == 0:
            if not ignore_zero:
                G.append(a)
                order += 1
            periodic = False
            break
        # check if we've reached something periodic
        elif a in G[:1]:
            break        
        G.append(a)
        order += 1
    return G, order, periodic