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Finite Group Z6


One of the two groups of Order 6 which, unlike $D_3$, is Abelian. It is also a Cyclic. It is isomorphic to $Z_2\otimes Z_3$. Examples include the Point Groups $C_6$ and $S_6$, the integers modulo 6 under addition, and the Modulo Multiplication Groups $M_7$, $M_9$, and $M_{14}$. The elements $A_i$ of the group satisfy ${A_i}^6=1$, where 1 is the Identity Element, three elements satisfy ${A_i}^3=1$, and two elements satisfy ${A_i}^2=1$. The Cycle Graph is shown above, and the Multiplication Table is given below.

$Z_6$ 1 $A$ $B$ $C$ $D$ $E$
1 1 $A$ $B$ $C$ $D$ $E$
$A$ $A$ 1 $E$ $D$ $B$ $C$
$B$ $B$ $E$ 1 $A$ $C$ $D$
$C$ $C$ $D$ $A$ 1 $E$ $B$
$D$ $D$ $B$ $C$ $E$ 1 $A$
$E$ $E$ $C$ $D$ $B$ $A$ 1

The Conjugacy Classes are $\{1\}$, $\{A\}$, $\{B\}$, $\{C\}$, $\{D\}$, and $\{E\}$.

See also Finite Group D3

© 1996-9 Eric W. Weisstein