Question

G={2,4,6,8} wrt multiplication under the modulo 10. Find its inverse law

Answer 1

$\textbf{Given:}$

$\mathsf{G=\{2,4,6,8\}\;with\;respect\;to\;multiplicaation\;modulo\;10}$

$\textbf{To find:}$

$\textsf{Inverse law}$

$\textbf{Solution:}$

$\textsf{First, we construct Cayley's table to find inverses of each element}$

$\begin{array}{|c|c|c|c|c|}\cline{1-5}{\times}_{10}&2&4&6&8\\\cline{1-5}2&4&8&2&6\\\cline{1-5}4&8&6&4&2\\\cline{1-5}6&2&4&6&8\\\cline{1-5}8&6&2&8&4\\\cline{1-5}\end{array}$

$\mathsf{Consider,}$

$\mathsf{2\;{\times}_{10}\;6=2}$

$\mathsf{4\;{\times}_{10}\;6=4}$

$\mathsf{6\;{\times}_{10}\;6=6}$

$\mathsf{8\;{\times}_{10}\;6=8}$

$\implies\textsf{6 is the identity element}$

$\mathsf{Also,}$

$\textsf{Inverse of 2 is 6}$

$\textsf{Inverse of 4 is 4}$

$\textsf{Inverse of 6 is 6}$

$\textsf{Inverse of 8 is 2}$

$\textsf{Hence, Inverse law is verified}$

$\textbf{Find more:}$

In a Group Z11 under multiplication operation the

inverse of 9 is​

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