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7. Let $G$ be a finite group, and let $A⊲⊲G$, $B⊲⊲G$ with $(|A|,|B|)=1$. Show that $⟨A,B⟩=A\times B$.

\begin{proof} Since $A\cap B=\{1\}$, it suffices to show $[A,B]=1$. Assume that the statement holds for group of order $<|G|$. Then if $⟨A,B⟩<G$, then $A,B$ are subnormal groups of $⟨A,B⟩$ and we finish the proof. So one can assume that $G=⟨A,B⟩$. If one of $A,B$ is normal in $G$, WLOG $A⊲G$, then $G=AB$ and $B$ is a subnormal Hall subgroup of $G$. In this case $B⊲G$ and so $G=A\times B$.

Now we assume that $A/⊲G$. We do induction on $|G:A|$. There exist subgroups $H,K⩽G$ such that $A⊲H⊲K$ and $A/⊲K$. Thus there exists $x\in K$ such that $A^x\ne A$. Since $A^x⊲H$, there is $A{A}^x=⟨A,A^x⟩$ and $(|A{A}^x|,|B|)=1$. Since $|G:A{A}^x|<|G:A|$, by induction hypothesis, we have $G=A{A}^x\times B$ and so $B⊲G$. Then $A$ is a subnormal Hall subgroup and so we finish the proof. \end{proof}

Remark. In fact, we can prove that if $A,B$ do not have common composition factor and $A,B⊲⊲G$, then $⟨A,B⟩=A\times B$. The proof is similar as above, see here.

10. Prove that $\mathrm{Aut}(A_5)\simeq S_5$.

\begin{proof} Let $a=(123)$, $b=(124)$, and $c=(125)$. Then $A_5=⟨a,b,c⟩$. For any $\mu \in \mathrm{Aut}(A_5)$, $a^{\mu },b^{\mu },c^{\mu }$ are still $3$-cycle. Note that

$$(a^{\mu }{b}^{\mu }{)}^2=(b^{\mu }{c}^{\mu }{)}^c=(b^{\mu }{a}^{\mu }{)}^2=(c^{\mu }{b}^{\mu }{)}^2=(a^{\mu }{c}^{\mu }{)}^2=1,$$

one can check $a^{\mu }=(ijk)$, $b^{\mu }=(ij\ell )$ and $c^{\mu }=(ijm)$, where $\{i,j,k,\ell ,m\}=\{1,2,3,4,5\}$. Define $d=\left(\begin{array}{ccccc}1& 2& 3& 4& 5\\ i& j& k& l& m\end{array}\right)$, then $a^{\mu }=a^d$, $b^{\mu }=b^d$ and $c^{\mu }=c^d$. Hence, $\mu$ is precisely the automorphism of $A_5$ induced by conjugation by $d$.

On the other hand, since $C_{S_5}(A_5)=1$, $S_5\to \mathrm{Aut}(A_5)$ is injective. Now we proved $\mathrm{Aut}(A_5)\simeq S_5$. \end{proof}

26. Let $G$ be a group, and let $H⩽\mathrm{Sym}(\Omega )$. For $G_1⩽G$ and $H_1⩽H$, there exists injective homomorphism from $G_1\wr H_1$ to $G\wr H$, and there exists $G_1{\wr }_r{H}_1\to G{\wr }_{r}H$, where $G{\wr }_{r}H\simeq G^{|H|}:H$.

\begin{proof} Consider “extending by the identity”. See here. \end{proof}

30. Define $P=Z_3\wr Z_3\wr Z_3⩽S_{27}$, which is a Sylow $3$-subgroup of $S_{27}$. Remark that $P$ is generated by $(123)$, $(147)(258)(369)$ and $(11019)(21120)\cdots (91827)$ (see 有限群导引, Theorem 5.6). Determine all minimal non-cyclic groups.

\begin{proof} By ^ub72lh, all minimal non-cyclic group is isomorphic to $C_3\times C_3$.

• later

\end{proof}