Abstract algebra library for Python (Work in progress)

I'm taking an abstract algebra course right now and an idea of creating a library for abstract algebra came to my mind. Any ideas on new features, how to improve etc. are welcome. And of course, any help on coding, optimization are gladly welcomed.

op - short for operation: Is an abstract method for defining the binary operation which is the operation the group is defined with.

elements: Is a property that returns the elements of the group

cartesianProduct: Returns the elements of the cartesian product of the group with another group

Zn is a cyclic group under addition and it's defined on the ring of integers modulo n. It is defined based on the Group class.

Examples:

Z14 = Zn(14)

Z13 = Zn(13)

Z14.elements # Returns [0, 1, ..., 13]

Z14.op(9, 7) # Returns 2 (9 + 7 = 16 % 14 = 2)

Z14.isGeneratedBy(3) # Returns true since gcd(14, 3) = 1

Z14.generateSubgroupBy(4) # Returns the generator of 4 in Z14, i.e. [4, 8, 12, 2, 6, 10, 0]

Z14.exists(5) # Checks if 5 exists in Z14

Z14.eye # Returns the identity element of Z14

Z14.n # Returns n

Z14.mod(21) # Returns 21 modulo 14, i.e 7

Z14.cartesianProduct(Z13) # Returns elements of the cartesian product

Zn* (Zn_coprime) is a group under multiplication and it's elements are integers less than and coprime with n. It is also defined based on the group class.

Examples:

Z14_coprime = Zn_coprime(14)

Z14_coprime.elements # [1, 3, 5, 9, 11, 13]

Z14_coprime.op(4, 5) # 6 (4*5 = 20 % 14 = 6)

Z14_coprime.mod(20) # 6

Z14_coprime.eye # 1

Z14_coprime.generateSubgroupBy(9) # [9, 11, 1]

Z13 = Zn(13)

Z14_coprime * Z13 # Returns the elements of cartesian product of the two group

Element class that represents a group element. It can be defined with a value and a group

Examples:

Z3 = Zn(3)

el_1 = Element(2, Z3)

el_2 = Element(1, Z3)

el_1 * el_2 # 0

el_1 > el_2 # True

el_2 > 2 # False

el_1 ** 2 # 1 (2 + 2 = 4 % 3 = 1)