Intereting Posts

Prove that finite dimensional $V$ is the direct sum of its generalized eigenspaces $V_\lambda$
How to calculate what matrix will transform specified points to other specified points
What is the “taxonomy” or “hierarchy” (partial ordering) of algebraic objects used to attempt to capture geometric intuition?
Trouble in understanding a proof of a theorem related to UFD.
Proof Strategy for a Dynamical System of Points on the Plane
Which books would you recommend about Recreational Mathematics?
Explain Carmichael's Function To A Novice
Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality
Does the sequence $(\sqrt{n} \cdot 1_{})_n$ converge weakly in $L^2$?
How to prove $\int_0^1\frac{x^3\arctan x}{(3-x^2)^2}\frac{\mathrm dx}{\sqrt{1-x^2}}=\frac{\pi\sqrt{2}}{192}\left(18-\pi-6\sqrt{3}\,\right)$?
“Why do I always get 1 when I keep hitting the square root button on my calculator?”
Define integral for $\gamma,\zeta(i) i\in\mathbb{N}$ and Stirling numbers of the first kind
How to generate REAL random numbers with some random and pseudo random
Divisibility and Pigeonhole principle
$f(x) – f'(x) = x^3 + 3x^2 + 3x +1; f(9) =?$

We know one of the presentation of $\mathbb Q_8$ is: $$\mathbb Q_8=\langle a,b,c|ab=c,bc=a,ca=b\rangle$$

and if we want to construct the semi-direct product of $\mathbb Q_8\rtimes\mathbb Z_3$; this can be carried out by defining a **proper** homomorphism, say $\phi$: $$\phi:=\mathbb Z_3\longrightarrow Aut(\mathbb Q_8)\cong\mathbb S_4$$ Usually, the groups which I had to examine, have been both cyclic, but this time one of them is the quaternion group, $\mathbb Q_8$. What I have learnt is to define a suitable homomorphism sending generators of groups to each other. So, here I should consider $\phi$ to send $x$ of order 3, as $\mathbb Z_3=\langle x\rangle$ to a correspondent element in $Aut(\mathbb Q_8)$.

My problem is to define a suitable homomorphism $\phi$ and then demonstrate an associated presentation of $\mathbb Q_8\rtimes_{\phi}\mathbb Z_3$. Thanks for the time you share.

- Why multiplicative group $\mathbb{Z}_n^*$ is not cyclic for $n = 2^k$ and $k \ge 3$
- Problem from Armstrong's book, “Groups and Symmetry”
- at least one element fixed by all the group
- Nilpotent groups are solvable
- What is the intersection of all Sylow $p$-subgroup's normalizer?
- counting the number of elements in a conjugacy class of $S_n$

- Primitive binary necklaces
- Needing help picturing the group idea.
- A question on $p$-groups, and order of its commutator subgroup.
- Show that all groups of order 48 are solvable
- Determining whether two groups are isomorphic
- Number of elements of order $p$ is a multiple of $p-1$ (finite group).
- Need for inverse in $1-1$ correspondence between left coset and right coset of a group
- $\mathbb{Z}$ is the symmetry group of what?
- Commutator subgroup of a free group
- Proper subgroups of non-cyclic p-group cannot be all cyclic?

Pick any element of order $3$ in $S_4$ to get a homomorphism $\varphi \colon ~ \mathbb{Z}_3 \to \mathrm{Aut}(\mathbb{Q}_8)$. The presentation of the semi-direct product is then given by

$$

\mathbb{Q}_8 \rtimes_\varphi \mathbb{Z}_3 = \langle a,b,c, x \mid ab = c, bc = a, ca = b, x^3 = 1, a^x = a^{\varphi(x)}, b^x = b^{\varphi(x)}, c^x = c^{\varphi(x)} \rangle,

$$

a disjoint union of presentations of the original groups plus the conjugation relations induced by the action $\varphi$.

Let $G=Q_8\rtimes \mathbb{Z}_3$, and suppose that action of $\mathbb{Z}_3$ on $Q_8$ (by conjugation) is non-trivial. Sicne $Q_8$ has three subgroups of order 4: $\langle i\rangle$, $\langle j\rangle$, $\langle k\rangle$; the conjugation action of $\mathbb{Z}_3$ on $Q_8$ will permute these subgroups. As $|\mathbb{Z}_3|=3$, orbit of a subgroup will have order 1 or 3; hence if one subgroup is fixed, then all subgroups will be fixed by $\mathbb{Z}_3$.

If one (hence all) subgroups fixed, then consider action of $\mathbb{Z}_3$ on $\langle i\rangle=\{1,-1,i,-i\}$ by conjugation. Since $-1$ is unique element of order 2 here, it will be fixed by $\mathbb{Z}_3$. Hence, $\mathbb{Z}_3$ will permute $\{i,-i\}$ by conjugation. But, again, orbit of $i$ should have order $1$ or $3$; the only possibility is that orbit should be singleton. We conclude that, if $\mathbb{Z}_3$ fixes $\langle i\rangle$, then it fixes this subgroups pointwise, and similarly, it will fix $\langle j\rangle$, $\langle k\rangle$ pointwise. Therefore, action of $\mathbb{Z}_3$ on $Q_8$ is trivial, a contradiction.

Hence, the non-trivial action of $\mathbb{Z}_3$ must permute the subgroups

$\langle i\rangle$, $\langle j\rangle$, $\langle k\rangle$ **cyclically**.

Now, we can easilt define a homomorphism you wanted: if $\mathbb{Z}_3=\langle z|z^3=1\rangle$, define

$z\mapsto \{ i\mapsto j, j\mapsto k, k\mapsto i\} $, i.e. $ z^{-1}.i.z=j, z^{-1}.j.z=k$, $z^{-1}.k.z=i$.

(**Remark**: this shows that there is only one non-trivial semidirect product of $Q_8$ by $\mathbb{Z}_3$; hence there is unique group $G$ such that $G=Q_8 \rtimes_{1} \mathbb{Z}_3$. The only such group is $SL(2,\mathbb{Z}_3)$.)

- Intuition behind the convolution of two functions
- Proving commutativity of addition for vector spaces
- Checking on some convergent series
- Are there an infinite number of prime numbers where removing any number of digits leaves a prime?
- Determination of the last three digits of $2014^{2014}$
- If a sub-C*-algebra does not contain the unit, is it contained in a proper ideal?
- A binary sequence graph
- How to solve $ 13x \equiv 1 ~ (\text{mod} ~ 17) $?
- Gamma Distribution out of sum of exponential random variables
- Local diffeomorphism is diffeomorphism provided one-to-one.
- If $f: \mathbb Q\to \mathbb Q$ is a homomorphism, prove that $f(x)=0$ for all $x\in\mathbb Q$ or $f(x)=x$ for all $x$ in $\mathbb Q$.
- How to do contour integral on a REAL function?
- What is the purpose of implication in discrete mathematics?
- How to prove a topologic space $X$ induced by a metric is compact if and only if it's sequentially compact?
- Proving that a polynomial is not solvable by radicals.