Klein-4 group

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The Klein-4 group (AKA: Viergruppe[1]), [ilmath]V_4=\{1,a,b,c\}[/ilmath] is a group on 4 elements with the operation [ilmath]*:V_4\times V_4\rightarrow V_4[/ilmath] defined as follows:

[ilmath] \begin{array}{|c|cccc|} \hline V_4 & 1 & a & b & c \\\hline 1 & 1 & a & b & c \\ a & a & 1 & c & b \\ b & b & c & 1 & a \\ c & c & b & a & a \\\hline \end{array}[/ilmath]
Operator tables for the Klien-4 group

Recall that to calculate the product say, [ilmath]a*b[/ilmath] we look at the row corresponding to [ilmath]a[/ilmath] and find the entry in the [ilmath]b[/ilmath] column, which is [ilmath]c[/ilmath] here (the same as [ilmath]b*a[/ilmath], so be sure you're doing it the correct way)

Claim: this is indeed a group.


  1. Abstract Algebra - Pierre Antoine Grillet