factor-groups.sage

from sage.all import Integers, factor


def example_40():
    Z5 = Integers(5)

    # get all elements
    # notice that we ignore 0, as this is a multiplicative group
    Z5_elems = []
    for e in range(1, Z5.order()):
        Z5_elems.append(Z5(e))
    Z5_elems = set(Z5_elems)
    print("Z5*   :", Z5_elems)

    # find largest prime factor
    order = Z5.order() - 1  # ignoring 0
    factorization = factor(order)
    lpf = max(factorization)[0]  # largest prime factor
    cf = order // lpf  # cofactor

    # clear cofactors
    Z5_2_elems = []
    for e in Z5_elems:
        Z5_2_elems.append(e**cf)
    Z5_2_elems = set(Z5_2_elems)
    print("Z5*[2]:", Z5_2_elems)


def exercise_40():
    Z6 = Integers(6)
    order = Z6.order()

    # find subgroup orders, which are given by the factorization of the order
    factorization = list(map(lambda x: x[0], factor(order)))
    subgroupOrders = [1, order]
    subgroupOrders.extend(factorization)
    subgroupOrders = sorted(subgroupOrders)

    # do co-factor clearing to compute subgroups
    factorGroups = {}
    for subgroupOrder in subgroupOrders:
        cf = order // subgroupOrder  # find cofactor
        Z6_f_elems = []
        for e in range(0, order):
            # notice that we use multiplication instead of exponent,
            # because this is an additive group
            Z6_f_elems.append(Z6(e) * cf)
        Z6_f_elems = set(Z6_f_elems)
        print("Z6[{}]  :{}".format(subgroupOrder, Z6_f_elems))
        factorGroups[subgroupOrder] = Z6_f_elems

    lpf = max(factorization)  # largest prime factor
    print("\nLargest Prime Order Subgroup:")
    print("Z6[{}]  :{}".format(lpf, factorGroups[lpf]))


def exercise_41():
    Zp = Integers(5)
    order = Zp.order()

    # clear co-factor for subgroup order 2
    so = 2
    cf = order // so

    sg_elems = []
    g_elems = []
    for e in range(1, order):
        g_elems.append(Zp(e))
        sg_elems.append(Zp(e) ** cf)
    sg_elems = set(sg_elems)
    g_elems = set(g_elems)

    print(g_elems)
    print(sg_elems)

    for g in sg_elems:
        powers = []
        for e in range(1, order):
            powers.append(g**e)
        print("{}: {}".format(g, powers))


if __name__ == "__main__":
    exercise_41()