The normalizer problem for finite groups having normal $2$-complements

1.   Introduction

Throughout this paper, $H$ denotes a finite group and $q$ is a prime. Denote the integral group ring of $H$ over $\mathbb{Z}$ by $\mathbb{Z}H$. The unit group of $\mathbb{Z}H$ is denoted by $\mathcal{U}(\mathbb{Z}H)$ denote and $\mathcal{Z}(\mathcal{U}(\mathbb{Z}H))$ to denote the center of $\mathcal{U}(\mathbb{Z}H)$. The normalizer problem asks whether $N_{\mathcal{U}(\mathbb{Z}H)}(H) = H\, \mathcal{Z}(\mathcal{U}(\mathbb{Z}H))$, where ${\rm N}_{\mathcal{U}(\mathbb{Z}H)}(H)$ is the normalizer of $H$ in $\mathcal{U}(\mathbb{Z}H)$ (see ^[1], Problem 43). If the equation holds, then it is said that $H$ has the normalizer property. Historically, it has been proven that the normalizer property holds for finite nilpotent groups in Coleman ^[2]. Jackowski and Marciniak ^[3] proved that the finite group having a normal Sylow $2$-subgroup has the normalizer property. In particular, groups of odd order have the normalizer property. A few years later, Mazur ^[4,5,6] found that the well-known isomorphism problem of the integral group ring and the normalizer problem are closely connected. Based on Mazur's observation, among other things, a first counterexample to the normalizer problem was found by Hertweck ^[7], and later a first counterexample to the isomorphism problem. However, investigating which groups have the normalizer property is still an interesting problem. Recently, a number of related works on this subject were published, see ^{[8,9,10,11,12]}.

To explain our results in detail, as in ^[10], some kinds of automorphisms of finite groups are defined in the following:

Let $\theta\in {\rm Aut}(H)$. We call $\theta$ a class-preserving automorphism if $\theta$ maps every element of $H$ to its conjugate. Denote by ${\rm Aut_{c}}(H)$ the class-preserving automorphism group of $H$.

Let $\theta\in {\rm Aut}(H)$. We call $\theta$ a Coleman automorphism if, for any $P\in {\rm Syl}(H)$, the restriction of $\theta$ to $P$ coincides with that of some inner automorphism of $H$. Denote by ${\rm Aut}_{\rm {Col}}(H)$ the Coleman automorphism group of $H$.

${\rm Aut}_{\mathbb{Z}}(H) = \{\theta_{u}\in {\rm Aut}(H)\mid x^{\theta_{u}} = u^{-1}xu, u\in {\rm N}_{\mathcal{U}(\mathbb{Z}H)}(H), x\in H\}\leq {\rm Aut}(H)$.

Write

Jackowski and Marciniak ^[3] showed that $N_{\mathcal{U}(\mathbb{Z}H)}(H) = H\, \mathcal{Z}(\mathcal{U}(\mathbb{Z}H))$ if and only if ${\rm Out}_{\mathbb{Z}}(H) = 1$. In addition, ${\rm Out}_{\mathbb{Z}}(H)\leq {\rm Out}_{c}(H)\cap {\rm Out_{Col}}(H)$, and ${\rm Out}_{\mathbb{Z}}(H)$ is an elementary abelian $2$-group (see ^[1]). Thus, ${\rm Out}_{\mathbb{Z}}(H) = 1$ is equivalent to the order of ${\rm Out_{c}}(H)\cap{\rm Out_{Col}}(H)$ being an odd number, i.e., $H$ has the normalizer property if and only if the order of ${\rm Out_{c}}(H)\cap{\rm Out_{Col}}(H)$ is an odd number.

In this paper, the normalizer problem of finite groups with normal $2$-complements is checked. Mazur ^[5] conjectured that finite groups having abelian Sylow $2$-subgroups have the normalizer property. He proved that the conjecture holds if Sylow $2$-subgroups have order $2$. Later, this result was generalized by Hertweck ^[11], who proved that $H$ has the normalizer property, provided that $H$ has a normal $2$-complement and $H$ has a cycli Sylow $2$-subgroup or an abelian of exponent at most $4$. Marciniak and Roggenkamp ^[13] showed that the normalizer property holds for metabelian groups having abelian Sylow $2$-subgroups. They also constructed a metabelian group $H = ({\rm C}_{2}^{4}\times {\rm C}_{3})\rtimes {\rm C}_{2}^{3}$ such that the order of ${\rm Out_{c}}(H)\cap {\rm Out_{Col}}(H)$ is an even number. This example illustrates that if the Sylow $2$-subgroup of $H$ is not abelian, then in general it is not the case that the order of ${\rm Out}_{c}(H)\cap {\rm Out_{Col}}(H)$ is an odd number. Our main results are the following:

Theorem 1.1. Let $M$ be a subgroup of $H$ such that $|H:M| = 2$. If $M = O_{2}(M)\times O_{2'}(M)$, then ${\rm Out}_{\mathbb{Z}}(H) = 1$, that is, $H$ has the normalizer property.

Theorem 1.2. Let $H = O_{2'}(H)\rtimes P$ be a semidirect product of a nilpotent normal $2$-complement $O_{2'}(H)$ by an abelian Sylow $2$-subgroup $P$. Then the order of ${\rm Out_{c}}(H)\cap {\rm Out_{Col}}(H)$ is an odd number. In particular, $H$ has the normalizer property.

Theorem 1.3. Let $H = O_{2'}(H)\rtimes Q$ be a semidirect product of a normal $2$-complement $O_{2'}(H)$ by a Hamilton Sylow $2$-subgroup $Q$. Then ${\rm Out}_{\mathbb{Z}}(H) = 1$.

Let $M\leq H$ or $M\unlhd H$, and $\sigma\in {\rm Aut}(H)$. If $\sigma$ fixes $M$ or $H/M$, which will be denoted by $\sigma|_{M}$ or $\sigma|_{H/M}$, respectively. For a $h\in H$, we use ${\rm conj}(h)$ to denote the inner automorphism induced by $h$. For any $p||H|$, denote by $O_{p}(H)$ the largest normal $p$-subgroup of $H$ and $O_{p'}(H)$ the largest normal $p'$-subgroup of $H$. The other notation is standard; refer to ^[10,14].

2.   Preliminaries

Lemma 2.1. ^[11] Assume that $H$ has cyclic Sylow $2$-subgroups. Then the order of ${\rm Out_{c}}(H)\cap {\rm Out_{Col}}(H)$ is an odd number.

Lemma 2.2. ^[11] Let $K$ be a normal $2$-complement of $H$. Suppose that $P\in {\rm Syl_{2}}(H)$, and $D\trianglelefteq P$ such that ${\rm exp}(P/D)\leqslant 4$. If $P/D$ is abelian and $\theta\in {\rm Aut}_{\mathbb{Z}}(H)$, then there is $\tau\in {\rm Inn}(H)$ such that $\theta\tau|_{KD}\in {\rm Aut_{Col}}(KD)$.

Lemma 2.3. ^[10] Suppose that $M\unlhd H$. Suppose that $q$ does not divide $|H/M|$. Then we have the following:

$\rm (1)$ If $\theta\in {\rm Aut_{c}}(H)$ is a $q$-element, then $\theta|_{M}\in {\rm Aut_{c}}(M)$;

$\rm (2)$ If $\theta\in {\rm Aut_{Col}}(H)$ is a $q$-element, then $\theta|_{M}\in {\rm Aut_{Col}}(M)$;

$\rm (3)$ If ${\rm Out_{c}}(M)$ or ${\rm Out_{Col}}(M)$ is a $q'$-group, then ${\rm Out_{c}}(H)$ or ${\rm Out_{Col}}(H)$ is also a $q'$-group.

Lemma 2.4. Let $M\leq H$, and let $\eta\in {\rm Aut}(H)$ be a $q$-element. If $\eta|_{M} = {\rm conj}(h)|_{M}$ for some $h\in H$, then there exists a $\delta\in {\rm Inn}(H)$ with the property that $\eta\delta|_{M} = id|_{M}$ and $o(\eta\delta)$ is still a power of the prime $q$.

Proof. Set $o(\eta) = q^{i}$ for some positive integer $i$. Then $\eta{\rm conj}(h^{-1})|_{M} = id|_{M}$. Let $j$ be a positive integer satisfying $(\eta{\rm conj}(h^{-1}))^{j}$ is the $q$-part of $\eta{\rm conj}(h^{-1})$, where $(j, q) = 1$. Thus, there are integers $s$ and $t$ such that $sj+tq^{i} = 1$. Obviously, the order of $(\eta{\rm conj}(h^{-1}))^{sj}$ is a power of $q$, and $(\eta{\rm conj}(h^{-1}))^{sj}|_{M} = id|_{M}$. Since ${\rm Inn}(H)\trianglelefteq {\rm Aut}(H)$, then there is a $\delta\in {\rm Inn}(H)$ such that $(\eta{\rm conj}(h^{-1}))^{sj} = \eta^{sj}\delta = \eta^{1-tq^{i}}\delta = \eta\delta$. We are done.

Lemma 2.5. Let $\theta\in {\rm Aut}(H)$ be a $q$-element. Suppose that $M\leq H$, such that $\theta$ fixes $M$. If $\theta|_{M} = {\rm conj}(h)|_{M}$ for some $h\in H$, then we may find a $q$-element $t\in H$ satisfying $\theta|_{M} = {\rm conj}(t)|_{M}$.

Proof. Set $o(\theta) = q^{i}$, $o(h) = q^{j}t$, and $(q, t) = 1$, where $i, j$, and $t$ are positive integers. Set $k: = max\{i, j\}$. Since $(q^{k}, t) = 1$, then there are integers $u$ and $v$ satisfying $uq^{k}+vt = 1$. Write $t = h^{vt}$. Consequently, $t$ is a $q$-element. For any $m\in M$, we have $m = m^{\theta^{uq^{{k}}}} = m^{h^{uq^{{k}}}}$, it follows that $m^{\theta} = m^{h} = m^{h^{uq^{k}+vt}} = (m^{h^{uq^{k}}})^{h^{vt}} = m^{h^{vt}} = m^{t}$, i.e., $\theta|_{M} = {\rm conj}(t)|_{M}$. We are done.

Lemma 2.6. ^[15] Let $\eta\in {\rm Aut}(H)$ be a $q$-element. Suppose that $M\unlhd H$, such that $\eta|_{M} = id|_{M}$, and that $\eta|_{H/M} = id|_{H/M}$. Then $\eta|_{H/O_{q}(Z(M))} = id|_{H/O_{q}(Z(M))}$. Further, if $\eta|_{R} = id|_{R}$ for some $R\in {\rm Syl_{q}}(H)$, then $\eta\in {\rm Inn}(H)$.

Lemma 2.7. ^[12] Assume that $Q$ is a $q$-group and $\bar{Q} = Q/\Phi(Q)$, where $\Phi(Q)$ is the Frattini subgroup of $Q$. Suppose that $B$ is an abelian $q'$-group such that $B$ acts on $Q$. Then one can find some $x\in Q$ satisfying $C_{B}(Q) = C_{B}(x) = C_{B}(\bar{Q}) = C_{B}({\bar x})$.

Lemma 2.8. ^[16] Let $\pi(H)$ and $\pi({\rm Aut_{Col}}(H))$ be the sets of prime divisors of $|H|$ and $|{\rm Aut_{Col}}(H)|$, respectively. Then $\pi({\rm Aut_{Col}}(H))\subseteq \pi(H)$.

Lemma 2.9. Let $\theta\in {\rm Aut}(H)$, and $M\unlhd H$. Then we have the following:

$\rm (1)$ If $\theta\in {\rm Aut_{c}}(H)$, then $\theta$ fixes $M$, and $\theta|_{H/M}\in {\rm Aut_{c}}(H/M)$;

$\rm (2)$ If $\theta\in {\rm Aut_{Col}}(H)$, then $\theta$ fixes $M$, and $\theta|_{H/M}\in {\rm Aut_{Col}}(H/M)$.

Proof. These proofs are obvious, so we omit them.

Lemma 2.10. ^[8] Let $v\in {\rm N}_{\mathcal{U}(\mathbb{Z}H)}(H)$, $M\unlhd H$, and let $Q$ be a $q$-subgroup of $H$. Assume that $v^{\eta} = Mh\in H/M$ for some $h\in H$, where $\eta: \mathbb{Z}H\rightarrow \mathbb{Z}(H/M)$ is the natural homomorphism. Then one can find some $m\in M$ such that $v^{-1}yv = (mh)^{-1}y(mh)$ for all $y\in Q$.

3.   Proof of The Theorems

Theorem 3.1. Let $M$ be a subgroup of $H$ satisfying $|H:M| = 2$. If $M = O_{2}(M)\times O_{2'}(M)$, then ${\rm Out}_{\mathbb{Z}}(H) = 1$, that is, $H$ has the normalizer property.

Proof. Let $\rho\in {\rm Aut}_{\mathbb{Z}}(H)$, and we will show that $\rho\in {\rm Inn}(H)$. If $O_{2}(M) = 1$, then Sylow $2$-subgroups of $H$ are cyclic. By Mazur's result, then ${\rm Out}_{\mathbb{Z}}(H) = 1$. Hereafter, we suppose that $O_{2}(M)\neq 1$. Let $Q\in {\rm Syl}_{2}(H)$. Then $O_{2}(M) = Q\cap M$. Since by assumption $|H:M| = 2$, it follows that $O_{2'}(M)$ is a normal $2$-complement $Q$ in $H$. By Lemma 2.2, then $\rho\gamma|_{M}\in {\rm Aut_{Col}}(M)$ for some $\gamma\in {\rm Inn}(H)$. Write $\beta = \rho\gamma$. Since ${\rm Out}_{\mathbb{Z}}(H)$ is an elementary abelian $2$-group, we may suppose that $o(\beta)$ is a power of $2$. So is $\beta|_{M}$. Note that $M = O_{2}(M)\times O_{2'}(M)$, where $O_{2}(M)\in {\rm Syl_{2}}(M)$, which implies that $\beta|_{M}\in {\rm Inn}(M)$ by Lemma 2.3(3). Thus, $\beta|_{M} = {\rm conj}(x)|_{M}$ for some $x\in M$. By Lemma 2.4, we obtain that $\beta\delta|_{M} = id|_{M}$ for some $\delta\in {\rm Inn}(H)$, and $o(\beta\delta)$ is still a power of $2$. By replacing $\beta$ with $\beta \delta$, we may suppose that

Since $H/M$ is cyclic, we obtain that

It is clear that $\beta$ acts on the set ${\rm Syl}_{2}(H)$. Since $|{\rm Syl}_{2}(H)|$ is an odd number and $\beta$ is of $2$-power order, one can find an $R\in {\rm Syl_{2}}(H)$ such that $R^{\beta} = R$. Note that ${\rm Aut}_{\mathbb{Z}}(H)\leq {\rm Aut_{Col}}(H)$, so $\beta|_{R} = {\rm conj}(g)|_{R}$ for some $g\in H$. By Lemma 2.5, we may suppose that $g$ is a $2$-element. It follows that $R^{\beta} = R = R^{g}$, i.e., $g\in N_{H}(R)$, which implies that $g\in R$. Note that $O_{2}(M) = M\cap R$. Then we have $\beta|_{O_{2}(M)} = {\rm conj}(g)|_{O_{2}(M)}$. However, $\beta|_{O_{2}(M)} = id|_{O_{2}(M)}$. Consequently, $x^{g} = x$ for all $x\in O_{2}(M)$, that is, $g\in C_{R}(O_{2}(M))$.

Case 1. Assume that $g$ is not in $O_{2}(M)$. Since $|R/O_{2}(M)| = |R/M\cap R| = |MR/M| = |H/M| = 2$, we obtain that $O_{2}(M)$ is a maximal subgroup of $R$. Thus, $R = < O_{2}(M), g >$. Since $g\in C_{R}(O_{2}(M))$, it implies that $g\in Z(R)$, and thus

Hence, by Lemma 2.6, Eqs (3.1)–(3.3) yield that $\beta\in {\rm Inn}(H)$, i.e., $\rho\in {\rm Inn}(H)$, as desired.

Case 2. Assume that $g\in O_{2}(M)$. Recall that $\beta|_{R} = {\rm conj}(g)|_{R}$. Write $\tau = \beta{\rm conj}(g^{-1})$. Then we have

Since $M = O_{2}(M)\times O_{2'}(M)$, $g\in C_{R}(O_{2}(M))$ and Eq (3.1), we have

In addition, by Eq (3.2) and $g\in O_{2}(M)\leq M$, we obtain that

Let $m$ be the $2'$-component of the order of $\tau$. Then $\tau^{m}$ is of $2$-power order and satisfies the following conditions: $\tau^{m}|_{R} = id|_{R}, \tau^{m}|_{M} = id_{M}, \tau^{m}|_{H/M} = id_{H/M}$. Thus, by Lemma 2.6, $\tau^{m}\in {\rm Inn}(H)$. Since ${\rm Out}_{\mathbb{Z}}(H)$ is an elementary abelian $2$-group, this implies that $\tau\in {\rm Inn}(H)$, i.e., $\beta{\rm conj}(g^{-1})\in {\rm Inn}(H)$. Hence $\rho\in {\rm Inn}(H)$. We are done.

The following results are immediate from Theorem 3.1, which generalizes the finite version of a result due to Li et al. (^[17], Theorem 2).

Corollary 3.2. Let $F(H)$ be the Fitting subgroup of $H$ satisfying $|H:F(H)| = 2$. Then ${\rm Out}_{\mathbb{Z}}(H) = 1$, that is, $H$ has the normalizer property.

Theorem 3.3. Let $H = O_{2'}(H)\rtimes P$ be a semidirect product of a nilpotent normal $2$-complement $O_{2'}(H)$ by an abelian Sylow $2$-subgroup $P$. Then the order of ${\rm Out_{c}}(H)\cap {\rm Out_{Col}}(H)$ is an odd number. In particular, $H$ has the normalizer property.

Proof. Let $\rho\in {\rm Aut_{c}}(H)\cap {\rm Aut_{Col}}(H)$ be of $2$-power order. Our goal is to prove that $\rho\in {\rm Inn}(H)$. Since $H = O_{2'}(H)\rtimes P$ is a semidirect product. First, we show that $\rho{\rm conj}(h^{-1})|_{O_{2'}(H)}\in {\rm Aut_{Col}}(O_{2'}(H))$ for some $h\in H$. Let $\pi(O_{2'}(H)) = \{q_{1}, q_{2}, \cdots, q_{r}\}$ and let $Q_{i}\in {\rm Syl}_{q_{i}}(O_{2'}(H))$, where $i = 1, 2, \cdots, r$. Then $O_{2'}(H) = Q_{1}\times Q_{2}\times \cdots \times Q_{r}$. Since $\rho\in {\rm Aut_{Col}}(H)$, we obtain that $\rho|_{Q_{i}} = {\rm conj}(t_{i})|_{Q_{i}}$ for some $t_{i}\in H$, where $i = 1, 2, \cdots, r$. In Lemma 2.5, we have

where $h_{i}\in H$ is a $2$-element. Since $Q_{i}\unlhd H$, thus $P$ acts on $Q_{i}$. By Lemma 2.7, $C_{P}(Q_{i}) = C_{P}(b_{i}) = C_{P}(\bar{b_{i}}) = C_{P}(\bar {Q_{i}})$ for some $b_{i}\in Q_{i}$. Set $b = b_{1}b_{2}\cdots b_{r}$. Since $\rho\in {\rm Aut_{c}}(H)$, then there is a $h\in H$ such that $b^{\rho} = b^{h}$, i.e., $(h^{-1}b_{1}h)\cdots(h^{-1}b_{r}h) = (h_{1}^{-1}b_{1}h_{1})\cdots (h_{r}^{-1}b_{r}h_{r})$. From this, we obtain $h^{-1}b_{i}h = h_{i}^{-1}b_{i}h_{i}$. It follows that

Since $H = O_{2'}(H)\rtimes P$, let $h_{i}h^{-1} = mk$, where $m\in O_{2'}(H)$ and $k\in P$. Next, we will show that $k\in C_{P}(Q_{i})$. Since $H$ acts on $\bar {Q_{i}}$, by Eq (3.8), we obtain that

On the other hand,

Since $\bar {Q_{i}}$ is abelian and $O_{2'}(H)$ is nilpotent, it follows that $[m, \bar{b_{i}}] = 1$. By (3.9) and (3.10), we obtain that $[k, \bar{b_{i}}] = 1$, that is, $k\in C_{P}(\bar{b_{i}}) = C_{P}(Q_{i})$. Thus, by (3.7), we imply that $\rho{\rm conj}(h^{-1})|_{Q_{i}} = {\rm conj}(m)|_{Q_{i}}$. This shows that $\rho{\rm conj}(h^{-1})|_{O_{2'}(H)}\in {\rm Aut_{Col}}(O_{2'}(H))$. Since $O_{2'}(H)$ is of odd order, by Lemma 2.8, $\rho{\rm conj}(h^{-1})|_{O_{2'}(H)} = id|_{O_{2'}(H)}$, that is, $\rho|_{O_{2'}(H)} = {\rm conj}(h)|_{O_{2'}(H)}$. In Lemma 2.4, let us set

Since $H/O_{2'}(H)$ is an abelian $2$-group. Then

By Lemma 2.6, Eqs (3.11) and (3.12) yield that $\rho|_{H/O_{2}(Z(O_{2'}(H)))} = id|_{H/O_{2}(Z(O_{2'}(H)))}$. Note that $|O_{2'}(H)|$ is an odd number. We obtain that $O_{2}(Z(O_{2'}(H))) = 1$, i.e., $\rho = id$. Hence $\rho\in {\rm Inn}(H)$. We are done.

A group $Q$ is called a Hamilton $2$-group if $Q = Q_{8}\times E$, where $Q_{8}$ is a quaternion group of order $8$ and $E$ is an elementary abelian $2$-group. Higman (^[1], Theorem 2.7) proved that the units of $\mathbb{Z}Q$ are trivial.

Theorem 3.4. Let $H = O_{2'}(H)\rtimes Q$ be a semidirect product of a normal $2$-complement $O_{2'}(H)$ by a Hamilton Sylow $2$-subgroup $Q$. Then ${\rm Out}_{\mathbb{Z}}(H) = 1$.

Proof. Let $\rho\in{\rm Aut}_{\mathbb{Z}}(H)$, and we will show that $\rho\in {\rm Inn}(H)$. By the definition of ${\rm Aut}_{\mathbb{Z}}(H)$, then there is a $v\in {\rm N}_{\mathcal{U}(\mathbb{Z}H)}(H)$ such that $h^{\rho} = v^{-1}hv$ for all $h\in H$. Let $\epsilon:\mathbb{Z}H\rightarrow \mathbb{Z}\, (\sum_{h\in H}r_{h}h\mapsto \sum_{h\in H}r_{h})$ be the augmentation map for $\mathbb{Z}H$ to $\mathbb{Z}$, where $r_{h}\in \mathbb{Z}$ for each $h\in H$. Then we have $\epsilon (v) = 1$ or $-1$ since $v$ is a unit of $\mathbb{Z}H$. Note that $\rho = {\rm conj}(v) = {\rm conj}(-v)$. Thus, we may suppose that $\epsilon(v) = 1$.

Consider the quotient $H/O_{2'}(H)$. We set $\bar{h}: = hO_{2'}(H)$ for any $h\in H$ and set $\bar{H}: = H/O_{2'}(H)$. Denote by

the natural homomorphism for $\mathbb{Z}H$ to $\mathbb{Z}\bar{H}$.

Since $v\in {\rm N}_{\mathcal{U}(\mathbb{Z}H)}(H)$, then $v^{\theta}\in {\rm N}_{\mathcal{U}(\mathbb{Z}\bar{H})}(\bar{H})$. By Lemma 2.9, $\rho$ fixes $O_{2'}(H)$ and $\rho$ induces an automorphism of $H/O_{2'}(H)$. Since $h^{\rho} = v^{-1}hv$ for any $h\in H$, then

Thus, $\rho|_{H/O_{2'}(H)}\in {\rm Aut}_{\mathbb{Z}}(H/O_{2'}(H))$. Since $H/O_{2'}(H)$ is a Hamilton $2$-group, by Higman's result, $\mathbb{Z}(H/O_{2'}(H))$ has only trivial units. Hence, one can find an element $\bar{g} = gO_{2'}(H)\in H/O_{2'}(H)$ satisfying $v^{\theta} = \bar{g}$. By Eq (3.14), we obtain that

Since $\rho{\rm conj}(g^{-1})\in {\rm Aut}_{\mathbb{Z}}(H)\subseteq {\rm Out_{c}}(H)\cap {\rm Out_{Col}}(H)$. By Lemma 2.9, $\rho{\rm conj}(g^{-1})|_{O_{2'}(H)}\in {\rm Aut}(O_{2'}(H))$. Next, we show that $\rho{\rm conj}(g^{-1})|_{O_{2'}(H)}\in {\rm Aut_{Col}}(O_{2'}(H))$. Let $p\in\pi(O_{2'}(H))$ and $P\in {\rm Syl}_{p}(O_{2'}(H))$. By Lemma 2.10, then $\rho{\rm conj}(g^{-1})|_{P} = {\rm conj}(n)|_{P}$ for some $n\in O_{2'}(H)$. Consequently, this shows that $\rho{\rm conj}(g^{-1})|_{O_{2'}(H)}\in {\rm Aut_{Col}}(O_{2'}(H))$. Write $\gamma: = \rho{\rm conj}((ng)^{-1})$. By Eq (3.15), we obtain that

Since ${\rm Out}_{\mathbb{Z}}(H)$ is an elementary abelian $2$-group, we may suppose that $o(\gamma)$ is a power of $2$. So is $\gamma|_{O_{2'}(H)}$. Note that $\gamma|_{O_{2'}(H)} = \rho{\rm conj}(g^{-1}){\rm conj}(n^{-1})|_{O_{2'}(H)}\in {\rm Aut_{Col}}(O_{2'}(H))$. By Lemma 2.8 and the fact that the order of $\gamma|_{O_{2'}(H)}$ is a power of $2$, we deduce that

Now, by Lemma 2.6, Eqs (3.16) and (3.17) one can see that $\gamma|_{H/O_{2}(Z(O_{2'}(H)))} = id|_{H/O_{2}(Z(O_{2'}(H)))}$. Since $O_{2'}(H)$ has odd order, we have $\gamma = id$. Thus, $\rho{\rm conj}((ng)^{-1}) = id$. Hence $\rho\in {\rm Inn}(H)$. We are done.

4.   Conclusions

In conclusion, we investigate the normalizer problem of finite groups with normal $2$-complements. We have proven that $H$ has the normalizer property, if $H$ is a semidirect product of a nilpotent normal $2$-complement by an abelian Sylow $2$-subgroup or $H$ is a semidirect product of a normal $2$-complement by a Hamilton Sylow $2$-subgroup. Additionally, we have proven that the normalizer property holds for finite groups with a nilpotent subgroup of index $2$.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Supported by the National Natural Science Foundation of China (11871292).

Conflict of interest

The authors declare there is no conflict of interest.