The influence of maximal quotient groups on the normalizer conjecture of integral group rings

Let N be a minimal normal subgroup of G such that Z⁢(G/N) has only trivial central units. Then the normalizer conjecture holds for Z⁢G.

Let N be a minimal normal subgroup of G such that G/N is a nilpotent group with a Dedekind Sylow 2-subgroup. Then the normalizer conjecture holds for Z⁢G.

Let G be a group and let v∈Z⁢G be an element with augmentation 1. Let φ be an automorphism of G such that v=g-1⁢v⁢gφ for all g∈G. Then for any Sylow subgroup P of G there is an element g in the support of v such that φ coincides with conjugation by g on P.

Theorem 2.2 (Krempa)

Let G be a group. Then OutZ⁢(G) is an elementary abelian 2-group.

Theorem 2.3 ([8, Theorem 3.6])

If G has a normal Sylow 2-subgroup, then OutZ⁢(G)=1.

Theorem 2.4 ([6, Theorem 14])

Let G be a simple group. Then there is a prime p∈π⁢(G) such that every p-central automorphism of G is inner. In particular, OutCol⁢(G)=1.

Theorem 2.5 ([9, Theorem 3.1])

Let G be a group with abelian Sylow 2-subgroups. Suppose that there exists a nilpotent normal subgroup N of G such that G/N has a normal Sylow 2-subgroup. Then every class-preserving Coleman automorphism of G of 2-power order is inner. In particular, OutZ⁢(G)=1.

Let G be a group with a normal subgroup N such that the central units of Z⁢(G/N) are trivial. Let ϕ∈AutZ⁢(G). Then there is an element b∈G such that for any Sylow subgroup P of G there exists an element a∈N with ϕ⁢conj⁡(b-1)|P=conj⁡(a)|P. Furthermore, ϕ⁢conj⁡(b-1)|G/N=id|G/N.

Proof.

Let H=G/N and let s:H→G be a section such that s⁢(eH)=eG, where eM is the identity element of a group M. Let u be a unit of NU⁢(ℤ⁢G)⁢(G) inducing ϕ via conjugation, i.e., gϕ=gu for all g∈G. We may assume that the augmentation ϵ⁢(u)=1. Then by [13, Lemma 2.1] the image u¯ of u in ℤ⁢H is an element t of H. Let b=s⁢(t), w=u⁢b-1 and let φ=ϕ⁢conj⁡(b-1). Then w¯=1 and w=g-1⁢w⁢gφ for any g∈G. We may write w=∑h∈Hwh⁢s⁢(h), where wh∈ℤ⁢N for each h∈H. Then we have 1=w¯=∑h∈Hϵ⁢(wh)⁢h. It follows that ϵ⁢(we)=1 and ϵ⁢(wh)=0 for all h≠e, where e=eH.

Let g∈G and let g¯ be its image in H. Then for any h∈H there exists ch∈N such that g-1⁢s⁢(h)⁢gφ=ch⁢s⁢(g¯-1⁢h⁢g¯). In particular, ce=g-1⁢gφ. Then we have

Note that g-1⁢wh⁢g⁢ch∈ℤ⁢N for any h∈H. It follows that wg¯-1⁢h⁢g¯=g-1⁢wh⁢g⁢ch for all h∈H. Taking h=e=eH, we get we=g-1⁢we⁢g⁢ce=g-1⁢we⁢gφ. Let P be a Sylow subgroup of G. Note that the support of we is contained in N. So by Theorem 2.1 there exists some a∈N such that φ coincides with conjugation by a on P. This means ϕ⁢conj⁡(b-1)|P=conj⁡(a)|P. Since φ=conj⁡(w) and w¯=1, the second assertion follows immediately. This completes the proof of Proposition 3.1. ∎

Let G be a group with a normal subgroup A of odd order such that the central units of Z⁢(G/A) are trivial. Then OutZ⁢(G)=1.

Let φ∈Aut⁢(G) be of p-power order, where p is a prime. Suppose that there is a normal subgroup N of G such that φ|N=id|N, φ|G/N=id|G/N. Then φ|G/Op⁢(Z⁢(N))=id|G/Op⁢(Z⁢(N)). Furthermore, if φ fixes element-wise a Sylow p-subgroup of G, then φ is an inner automorphism of G.

Proof of Theorem A.

Let G and N be as in Theorem A. Let ϕ∈Autℤ⁢(G). We have to show that ϕ∈Inn⁢(G). The proof is divided into two cases according to whether N is abelian or not.

Case 1: N is abelian. If N is of odd order, then the assertion follows from Corollary 3.2. Note that N is an elementary abelian p-group. So it remains to consider the case when N is of 2-power order. Take a Sylow 2-subgroup P of G. Then N≤P. By Proposition 3.1, there exists b∈G such that φ:=ϕ⁢conj⁡(b-1) acts as conjugation by some element x∈N on P and induces identity on G/N. Namely, we have φ|P=conj⁡(x)|P and φ|G/N=id|G/N. Write σ:=φ⁢conj⁡(x-1). Then we have σ|P=id|P and σ|G/N=id|G/N. Note that N is contained in P. So we also have σ|N=id|N. Since by Theorem 2.2 σ2 is inner, without loss of generality, we may assume that σ is of 2-power order. Applying Lemma 3.3, we have σ∈Inn⁢(G), yielding that ϕ∈Inn⁢(G).

Case 2: N is non-abelian. Since ϕ∈Autℤ⁢(G), it follows from Proposition 3.1 that there exists an element b∈G such that φ:=ϕ⁢conj⁡(b-1) acts as conjugation by some element in N on every Sylow subgroup of G. It follows that φ, when restricts to N, is a Coleman automorphism of N. Since N is a non-abelian minimal normal subgroup of G, it follows that N is a direct product of isomorphic non-abelian simple groups. It is known that OutCol⁢(⋅) is closed under taking direct products (see [3, Remark 4.3]). So by Theorem 2.4, OutCol⁢(N)=1 and thus φ|N=conj⁡(x)|N for some x∈N. In addition, by Proposition 3.1, we have φ|G/N=id|G/N. Write σ:=φ⁢conj⁡(x-1). Then we have σ|N=id|N and σ|G/N=id|G/N. As before, we may assume that σ is of 2-power order. Note that Z⁢(N)=1. So by Lemma 3.3, we have σ∈Inn⁢(G), implying ϕ∈Inn⁢(G). This completes the proof of Theorem A. ∎

Let G be an extension of a simple group by a group whose integral group ring has only trivial central units. Then the normalizer conjecture holds for Z⁢G. In particular, this is the case when G is an extension of a simple group by the symmetric group Sn of degree n.

Proof of Theorem B.

Let G,N be as in Theorem B. Write H:=G/N. Then by hypothesis, H is a nilpotent group with a Dedekind Sylow 2-subgroup, say P. Let σ∈Autℤ⁢(G). We have to show σ∈Inn⁢(G). The proof is divided into two cases according to whether P is abelian or Hamiltonian 2-group.

Case 1: P is abelian. The proof splits into two subcases according to whether N is abelian or not.

Subcase 1.1: N is abelian. Then N is an elementary abelian p-group for some prime p. If p=2, then G has a normal Sylow 2-subgroup and thus the assertion follows from Theorem 2.3. If p≠2, then G has an abelian Sylow 2-subgroup. Note that G is a nilpotent-by-nilpotent group. So the assertion follows from Theorem 2.5.

Subcase 1.2: N is non-abelian. As σ∈Autℤ⁢(G), it follows that σ|N∈Aut⁢(N). Let N=S×S×⋯×S, where S is a non-abelian simple group. By Theorem 2.4, there exists a prime q such that every q-central automorphism of S is inner. Let Q be a Sylow q-subgroup of N. Then there exists some element g∈G such that σ|Q=conj⁡(g)|Q. Write φ:=σ⁢conj⁡(g-1). Then φ|N∈Aut⁢(N). It follows that φ|N permutes on the set of all minimal normal subgroups of N. On the other hand, note that φ|Q=id|Q; in particular, φ|S∩Q=id|S∩Q for each S. We conclude that φ|N must fix every S, i.e., φ|S∈Aut⁢(S). Consequently, φ|S is a q-central automorphism of S and thus φ|S∈Inn⁢(S). As N is the direct product of copies of S, it follows that φ|N∈Inn⁢(N). Let x∈N such that φ|N=conj⁡(x)|N. Write ϕ:=φ⁢conj⁡(x-1). Then ϕ|N=id|N. Since H is nilpotent and ϕ|H∈Autℤ⁢(H), it follows from Theorem 2.1 that ϕ|H=conj⁡(h)|H for some h∈H. Note that ϕ∈Autℤ⁢(G). So, without loss of generality, we may assume that ϕ is of 2-power order and that h is of 2-power order. Since H is a nilpotent, it follows that h∈P. Keep in mind that in our case P is abelian. So actually ϕ|H=conj⁡(h)|H=id|H. Now by Lemma 3.3 we have ϕ|G/O2⁢(Z⁢(N))=id|G/O2⁢(Z⁢(N)). Note that in our case Z⁢(N)=1. So the previous equality yields that ϕ=id, implying σ∈Inn⁢(G), as desired.

Case 2: P is Hamiltonian. Let P=Q8×E, where E denotes an elementary abelian 2-group. The proof of this case splits into two subcases according to whether N is abelian or not.

Subcase 2.1: N is abelian. Then N is an elementary abelian p-group for some prime p. In the case when p=2, the group G has a normal Sylow 2-subgroup and thus the assertion follows from Theorem 2.3. It remains to consider the case when p≠2. Let L be the normal subgroup of G such that L/N=O⁢(H), where O⁢(H) denotes the maximal normal subgroup of H of odd order. Then L is of odd order and G/L≅G/N/L/N≅P. This shows that G may be regarded as an extension of an odd order group L by Q8×E. Since ℤ⁢(Q8×E) has only trivial units, it follows from Corollary 3.2 that the normalizer conjecture holds for ℤ⁢G.

Subcase 2.2: N is non-abelian. Denote by M the normal subgroup of G such that M/N=O⁢(H). We claim that OutCol⁢(M) is a 2′-group. Let ρ∈AutCol⁢(M) be of 2-power order. We have to show that ρ∈Inn⁢(M). As in the proof of Subcase 1.2 above, without loss of generality, we may assume that ρ|N=id|N. Since ρ∈AutCol⁢(M), it follows that ρ|M/N∈AutCol⁢(M/N). Note that M/N is of odd order. So, on the one hand, by [6, Proposition 1], ρ|M/N is of odd order. On the other hand, by hypothesis, ρ is of 2-power order, so ρ|M/N is of 2-power order. Consequently, we must have ρ|M/N=id|M/N. Now, applying Lemma 3.3, we obtain that ρ|M/O2⁢(Z⁢(N))=id|M/O2⁢(Z⁢(N)). Since N is a non-abelian minimal normal subgroup of G, it follows that Z⁢(N)=1. Hence the previous equality yields ρ=id. As ρ is arbitrary, OutCol⁢(M) is a 2′-group, as desired.

Note that G/M≅P=Q8×E. So ℤ⁢(G/M) has only trivial units. Thus by Proposition 3.1, there exists some b∈G such that for any Sylow subgroup Q of G there is s∈M with σ⁢conj⁡(b)|Q=conj⁡(s)|Q. In particular, this implies that σ⁢conj⁡(b)|M∈AutCol⁢(M). As before, we may assume that σ⁢conj⁡(b) is of 2-power order. Since OutCol⁢(M) is a 2′-group, there exists some element x∈M such that σ⁢conj⁡(b)|M=conj⁡(x)|M. Applying Proposition 3.1 again, we have σ⁢conj⁡(b)|G/M=id|G/M. Write γ=σ⁢conj⁡(b⁢x-1). Then we have γ|M=id|M and γ|G/M=id|G/M. Without loss of generality, we may assume that γ is of 2-power order. Then by Lemma 3.3, we have γ|G/O2⁢(Z⁢(M))=id|G/O2⁢(Z⁢(M)). It is clear that O2⁢(Z⁢(M))=1 since O2⁢(Z⁢(M)) is contained in N and N is a non-abelian minimal normal subgroup of G. Thus we actually have γ=id. Namely, σ=conj⁡(x⁢b-1)∈Inn⁢(G). The proof of Theorem B is finished. ∎

Let G be an extension of a simple group by a Dedekind group. Then the normalizer conjecture holds for Z⁢G.

Let N be a minimal normal subgroup of a group G such that G/N is an abelian group. Then OutCol⁢(G)=1. In particular, this is the case when G is an extension of a simple group by an abelian group.

Proof.

Let σ∈AutCol⁢(G). We have to show that σ∈Inn⁢(G). We first consider the case when N is abelian. In this case, N is an elementary abelian p-group for some prime p. Let P be the Sylow p-subgroup of G. Then by the definition of σ there exists some x∈G such that σ|P=conj⁡(x)|P. Without loss of generality, we may assume that σ|P=id|P. In particular, we have σ|N=id|N. Since G/N is abelian, it follows that σ|G/N=id|G/N. Thus o⁢(σ) divides the order N and hence σ is of p-power order. Applying Lemma 3.3, we have σ∈Inn⁢(G), as desired. It remains to consider the case when N is non-abelian. The proof for this case is similar to that of Subcase 1.2 in Theorem B. So we leave it to the reader. ∎

Suppose that the derived subgroup G′ is a minimal normal subgroup of G. Then OutCol⁢(G)=1. In particular, this is the case when G′ is a simple group.

Let N be a minimal normal subgroup of G such that the Sylow 2-subgroup of G/N is both Dedekind and a direct factor of G/N. Then the normalizer conjecture holds for Z⁢G.

In [10] Kimmerle proved that if G has no composition factor of order 2, then OutCol⁢(G) is a 2′-group (see [10, Proposition 3] and [6, Theorem 14]). In this spirit, we shall record the following result: if G has a chief series of length 2, then OutCol⁢(G)=1; in particular, this is the case when G has a composition series of length 2. In fact, if all non-trivial normal subgroups of G have the same order, then the assertion follows from [16, Theorem B]. Otherwise, G has two minimal normal subgroups N and M say, such that |N|<|M|. Since G has a chief series of length 2, it follows that {N,G/N}={M,G/M} (up to isomorphism). So |N|=|G/M| and N∩M=1. We obtain that G=M×N. Since OutCol⁢(M)=OutCol⁢(N)=1 and OutCol⁢(⋅) is closed under taking direct products (see [3, Remark 4.3]), it follows that OutCol⁢(G)=1.