Showing ${\rm Aut}(Q_{2^n})\cong{\rm Hol}(\Bbb Z_{2^{n-1}})$ for $n>3$

Question

This is part of Exercise 5.3.4 of Robinson's, "A Course in the Theory of Groups (Second Edition)". According to this search, it is new to MSE.

The second part is here:

• Automorphism group of the quaternion group

For previous questions of mine on generalised quaternion groups, see here:

• Find the upper central series of $Q_{2^n}$.

• Showing ${\rm Aut}(D_{2^n})\cong{\rm Aut}(Q_{2^n})$ for $n\ge 4$.

The Details:

On page 37 of the book,

Let $\lambda:G\to {\rm Sym}\, G$ and $\rho:G\to{\rm Sym}\, G$ be the left and right regular representations of a group $G$. Then $G^\lambda$ and $G^\rho$ are subgroups of ${\rm Sym}\, G$, as is ${\rm Aut}(G)$. Now $g^\lambda g^\rho$ maps $x$ to $g^{-1}xg$, so $g^\lambda g^\rho$ is just $g^\tau$, the inner automorphism induced by $g$. Consequently

$$\langle G^\lambda, {\rm Aut}(G)\rangle =\langle G^\rho, {\rm Aut}(G)\rangle;$$

this subgroup of ${\rm Sym}\, G$ is known as the holomorph of the group $G$,

$${\rm Hol}\, G.$$

Elsewhere in the book, we have:

$$D_{2^n}\cong \langle r,s\mid r^{2^{n-1}}, s^2, srs=r^{-1}\rangle$$

is the dihedral group of order $2^n$ and

$$Q_{2^n}\cong\langle x,y\mid x^{2^{n-1}}, y^2=x^{2^{n-2}}, y^{-1}xy=x^{-1}\rangle$$

is the generalised quaternion group of order $2^n$ (defined for $n\ge 3$).

The Question:

Prove that ${\rm Aut}(Q_{2^n})\cong {\rm Hol}(\Bbb Z_{2^{n-1}})$ if $n>3$.

Thoughts:

The answers to my previous question (linked to above) establish that

$${\rm Aut}(D_{2^n})\cong{\rm Aut}(Q_{2^n})$$

for $n>3$. It might help.

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By definition, we have

$${\rm Hol}(\Bbb Z_{2^{n-1}})=\langle \Bbb Z_{2^{n-1}}^\lambda, {\rm Aut}(\Bbb Z_{2^{n-1}})\rangle.$$

Here $\Bbb Z_{2^{n-1}}^\lambda$ is the image of $\Bbb Z_{2^{n-1}}$ under $\lambda$, i.e.,

$$\Bbb Z_{2^{n-1}}^\lambda=\{ g^\lambda: x\mapsto g^{-1}x\mid g\in \Bbb Z_{2^{n-1}}\}$$

and it is well-known that

$${\rm Aut}(\Bbb Z_{2^{n-1}})\cong U(2^{n-1}),$$

where $U(m)$ is the group of units modulo $m$. (See Theorem 6.5 of Gallian's "Contemporary Abstract Algebra (Eighth Edition)".)

That's all I have there.

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In GroupNames, we have

• ${\rm Aut}(Q_{2^4})\cong \langle a,b,c\mid a^8=b^2=c^2=1, bab=a^3, cac=a^5, bc=cb\rangle\cong \Bbb Z_8\rtimes\Bbb Z_2^2$.

• ${\rm Aut}(Q_{2^5})\cong\langle a,b,c\mid a^{16}=b^2=c^4=1, bab=a^{-1}, cac^{-1}=a^5, bc=cb\rangle\cong D_{32}\rtimes \Bbb Z_4$.

• $\lvert{\rm Aut}(Q_{2^6})\rvert=512$.

• $\lvert{\rm Aut}(Q_{2^7})\rvert=2,048$.

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Please help :)