Why is this quotient group not cyclic?

Asked Apr 20 '21 at 10:07

Active Apr 20 '21 at 10:18

Viewed 33 times

Consider $D_4$ generated by the "reflection" element, $\sigma$, and the "rotation" element $\rho$.
$D_4 = \{\mathbb{1}, \rho , \rho^2 , \rho^3 , \sigma, \sigma \rho , \sigma \rho^2 , \sigma \rho^3\}$

Now consider the (cyclic) group of order $2$, $\mathbb{Z}_2[\sigma] = \{\mathbb{1} , \sigma \}$

Then $D_4 / \mathbb{Z}_2[\sigma] = \{ \{ \mathbb{1},\sigma\} , \{ \rho ,\sigma\rho\} , \{ \rho^2,\sigma\rho^2\} , \{\rho^3,\sigma\rho^3 \} \} = \{\{ \mathbb{1},\sigma\} , \rho\{ \mathbb{1},\sigma\} , \rho^2\{ \mathbb{1},\sigma\}, \rho^3\{ \mathbb{1},\sigma\} \} $

Which is surely cyclic? But I have been told that it is not. Could anyone clear something up for me

Thanks :)

edited Apr 20 '21 at 10:18

asked Apr 20 '21 at 10:07

OskarSzarowicz

1,917

Why should the quotient group be cyclic? You forgot to use the quotient group multiplication. – Dietrich Burde Apr 20 '21 at 10:16
Yes I have now added that element in but still the group should be cyclic @DietrichBurde – OskarSzarowicz Apr 20 '21 at 10:19
I had mistyped but have edited it in now @Bernard – OskarSzarowicz Apr 20 '21 at 10:19
See the answer by Tanner in the duplicate "the quotient group of order 4 is isomorphic to the Klein four group, in which all three non-identity elements have order 2, which is not isomorphic to the cyclic group of order 4". – Dietrich Burde Apr 20 '21 at 10:23
1

Your subgroup ${1,\sigma}$ isn't normal. So the quotient isn't a group. – Apr 20 '21 at 10:40

Why is this quotient group not cyclic?

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