Prove that $Q_8$ has a subgroup $H$ of order $2$ and $H$ is normal in $Q_8$

Asked Oct 07 '16 at 10:54

Active Oct 07 '16 at 10:54

Viewed 157 times

Attempt

We have Quantemion group $Q_8=\{\pm 1, \pm i, \pm j,\pm k\}$. Its subgroups are $\{1\}, \{-1,1\}, \{-1,1,-i,i\}, \{-1,1,-j,j\}, \{-1,1,-k,k\}$, and $\{-1,1,-i,i,-j,j,-k,k\}$ of which $H=\{-1,1\}$ is a subgroup of order 2.

Question: 1. Is there any other subgroup of order 2. 2. How to show that this subgroup of order 2 is normal in $Q_8$?

Please help.

asked Oct 07 '16 at 10:54

user1942348

3,871

There is only one subgroup of order $2$, see here. – Dietrich Burde Oct 07 '16 at 11:00
@DietrichBurde How to show that it is normal? – user1942348 Oct 07 '16 at 11:01
Because it is equal to the group center - see the duplicate above. – Dietrich Burde Oct 07 '16 at 11:01
How to show that "it is equal to the group center" – user1942348 Oct 07 '16 at 11:02
1

Only $\pm 1$ commute with $i,j,k$. See also here, where Johannes Huisman computes the center of $Q_8$. – Dietrich Burde Oct 07 '16 at 11:04

Prove that $Q_8$ has a subgroup $H$ of order $2$ and $H$ is normal in $Q_8$

0 Answers0