Let G=S3andH={(1)(2)(3),(12)(3)}. Find the left cosets of H∈G.

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doplovif · Accepted Answer

Given&amp;nbsp;G=S3&amp;nbsp;and&amp;nbsp;H={(1)(2)(3),(12)(3)}={(I),(12)}&amp;nbsp;Number of element&amp;nbsp;∈H=2&amp;nbsp;Left coset&amp;nbsp;If H is a subgroup of G, then G is a group. Then the subset&amp;nbsp;aH={ah∣h∈H}⊆G&amp;nbsp;is the left coset of H contaning a.&amp;nbsp;Elementing&amp;nbsp;∈G.&amp;nbsp;S3={(I),(12),(13)(23)(123)(132)}&amp;nbsp;6 total elements are present.Here&amp;nbsp;Total number of left coset= (Number of elements in G)/(Number of elements&amp;nbsp;∈H)=62=3&amp;nbsp;Find the cosets as&amp;nbsp;S3={(I),(12),(13)(23)(123)(132)}&amp;nbsp;H={(I),(12)}&amp;nbsp;(I)H=H&amp;nbsp;(12)H={(12),(1)(2)(3)}&amp;nbsp;={(12),(I)}=(I)H=H&amp;nbsp;(13)H={(13),(13)(12)}&amp;nbsp;={(13),(123)}&amp;nbsp;(23)H={(23),(23)(12)}&amp;nbsp;={(23,(132)}&amp;nbsp;&amp;nbsp;(132)H={(23,(132)}=(23)H&amp;nbsp;&amp;nbsp;(123)H={(13),(123)}=(13)H&amp;nbsp;Therefore, three different cosets are&amp;nbsp;{(I)H,(23)H,(13)H}

Let G=S_3 and H={ (1)(2)(3),(12)(3)}. Find the left cosets of H in G.

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