Answer to Question #120559 in Abstract Algebra for inv

Question

Build a set of operation tables for group G with orders from 1, 2, 3 and 4 using the elements of a, b, c, and e as the identity element.

Answer 1

Group of order 1 $\equiv \{e\}$ .

Group of order 2 $\equiv \Z_2 = \{e,a\}$ where $a$ is element of order 2.

Group of order 3 $\equiv \Z_3 = \{e,b,b^2\}$ where $b$ are element of order 3.

Group of order 4 is either isomorphic to $\Z_4$ or $\Z_2 \times \Z_2$

where $\Z_4 = \{e,c,c^2,c^3\}$: $c$ is an element of order 4

and $\Z_2 \times \Z_2 = \{e, f,g,h\}$: $f,g,h$ are elements of order 2.

Composition table is as follows: