Introduction to Group Theory

Binary Operations

Page Info

On this page, you can create your own group and check to see whether it is well-formed (meaning it fits the four axioms described on the previous page).

First, fill out each of the "elements" fields to create a set. You can then choose a binary operation (multiplication or addition, with the choice to take the modulo). You can then check to see if the group has closure, associativity, the identity element (which is not always what you'd expect; try the set 2,4,6,8 $$\pmod{10}$$), and the inverse element. The size of the set ranges from 2 to 8, so you have to check the size of the group, then fill out the textboxes from left to right based on the size you selected. Then click "Create Group".

Close

Group Elements:

Operator:
$$\times$$ (multiplication)

Modulo

Test each of the four axioms:

Does your group satisfy all four?

Table size:
2x2
3x3
4x4
5x5
6x6
7x7
8x8