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16E

Expert-verifiedFound in: Page 246

Book edition
3rd

Author(s)
Thomas W Hungerford, David Leep

Pages
608 pages

ISBN
9781111569624

**Suppose G is a cyclic group ${<a>}$ and ${\left|a\right|}$=15 . If role="math" localid="1651649969961" ${\mathit{K}}{\mathbf{}}{\mathbf{=}}{<{a}^{3}>}$, list all the distinct cosets of K in G .**

The distinct cosets of *K* in *G *are *K e, K a, K a ^{2}* .

Given that *G* is a cyclic group $>">a$ of order 15 which are {1, a ,a^{2},a^{3},a^{4},a^{5},a^{6},a^{7}a,^{8},a^{9},a^{10},a^{11},a^{12},a^{13},a^{14}} .

Let *K* = $>">{a}^{3}$ be the subgroup of *G* then, the cosets of *K* can be separated as *Ke*{1, a^{3}, ^{5},a^{6 },a^{9 },a^{12 }}, *K**a ^{2} = {a^{2},a^{5},a,^{8},a^{11} ,a^{14}} . }* and .

Therefore, the distinct cosets of *K* in *G* are *Ke, Ka, Ka*^{2} .

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