13
$\begingroup$

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:

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

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.


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.


In GroupNames, we have

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

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

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

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


Please help :)

$\endgroup$
5
  • 5
    $\begingroup$ The previous question basically describes all the automorphisms and gives you the solution to this question. Look at e.g. the automorphisms $\alpha(a,b)$ in the answer by Derek Holt. Calculate $\alpha(a,b) \circ \alpha(a',b')$. This is equal to $\alpha(a'',b'')$ for some $a''$ and $b''$, what are $a''$ and $b''$? $\endgroup$
    – spin
    Commented Jan 8, 2022 at 8:33
  • 5
    $\begingroup$ Yes I agree, it follows easily from my previous answer. The elements of the automorphism group with $a=1$ form a normal subgroup that is isomorphic to $C_{2^{n-1}}$, and the elements with $b=0$ form a complementary subgroup isomorphic to ${\rm Aut}(C_{2^{n-1}})$. $\endgroup$
    – Derek Holt
    Commented Jan 8, 2022 at 9:23
  • 22
    $\begingroup$ @markvs I don't think it is possible to copyright exercise to such an extent. This site would not have much content if that were the case, I think :-) $\endgroup$ Commented Jan 8, 2022 at 9:40
  • 4
    $\begingroup$ Please let us use the meta thread for any further debate about copyright, etc. $\endgroup$
    – Pedro
    Commented Jan 9, 2022 at 14:10
  • 5
    $\begingroup$ Here is the thread mentioned by @PedroTamaroff. $\endgroup$
    – Shaun
    Commented Jan 9, 2022 at 19:24

0

You must log in to answer this question.