Solution to a problem about central subgroups

In the paper “Good involutions of conjugation subquandles” [LT25], I posed a question about involutions in certain central subgroups of nonabelian groups. This very short blogpost answers that question in the positive. Our example is the central product \((D_4\times C_4)/C_2\), which has order 16.

1. Introduction

The purpose of this blogpost is to give a positive answer to Problem 7.3 in my paper [LT25], which asks about a certain central subgroup \(A(G)\) of an arbitrary group \(G\). Namely, let \(A(G)\) be the set of central elements \(a\in Z(G)\) that satisfy the following property: There exist a positive integer \(n\in\mathbb{Z}^+\) and elements \(g_1,\dots,g_{2n}\in G\) such that \[a=g_1g_2^{-1}g_3g_4^{-1}\cdots g_{2n-1}g_{2n}^{-1} = g_1^{-1}g_2g_3^{-1}g_4\cdots g_{2n-1}^{-1}g_{2n}.\] On the other hand, let \[T(G):=\{z\in Z(G)\mid z^2=1\}\] be the subgroup of central involutions of \(G\). Note that \(A(G)\) is a subgroup of \(Z(G)\), and \(T(G)\) is a subgroup of \(A(G)\). Problem 7.3 asks whether there exists a group \(G\) such that \(A(G)\neq T(G)\). Note that if we require that \(Z(G)\in\{G,T\}\), then the problem has a negative answer.

The definition of \(A(G)\) may seem artificial, so let us discuss its motivation briefly. The aforementioned paper studies algebraic structures called symmetric quandles or quandles with good involutions, which are used to study surface-knots in \(\mathbb{R}^4\). A question of Taniguchi [YT23] seeks a characterization of good involutions of a given quandle. My paper studied this question for all quandles that embed into conjugation quandles \(\mathrm{Conj}(K)\) of groups \(K\), including all core quandles \(\mathrm{Core}(G)\) of groups \(G\).

To study good involutions of \(\mathrm{Core}(G)\) (see [LT25, Thm. 7.11]), we pass to the conjugation quandle of the group \(K:=(G\times G)\rtimes\mathbb{Z}/2\mathbb{Z}\), where \(\mathbb{Z}/2\mathbb{Z}\) acts on \(G\times G\) by swapping coordinates. It turns out that \(A(G)\) is isomorphic to the center of the subgroup of \(K\) generated by elements of the form \((g,g^{-1},1)\); see [LT25, Prop. 7.8]. We exploit this fact to determine all good involutions of \(\mathrm{Core}(G)\) in terms of functions into \(A(G)\), and then we refine our result by replacing \(A(G)\) with \(T(G)\). Therefore, it were true that \(A(G)=T(G)\), then the proof could be streamlined. The result of this blogpost shows that this is not possible in general.

2. The example

Our example is the following group of order 16; see [Cent] for more information about this group. Let \(G:=(D_4\times C_4)/C_2\) be the central product of the dihedral group \(D_4\) of order eight and the cyclic group \(C_4\) of order four over a common cyclic central subgroup of order two: \[G\cong\langle r,s,x\mid r^4=s^2=x^4=1,\; r^2=x^2,\; srs=r^{-1},\; rx=xr,\; sx=xs \rangle.\] Here, \(\langle r,s\rangle\cong D_4\) and \(Z(G)=\langle x\rangle\cong C_4\).

In the above presentation of \(G\), we have \(x\notin T(G)=\langle x^2\rangle\cong C_2\). We claim that \(x\in A(G)\). Indeed, let \[g_1:= xr^3,\quad g_2:= s,\quad g_3:= 1,\quad g_4:= sr.\] Then \[ g_1g_2^{-1}g_3g_4^{-1}= (xr^3)(s)(sr)=x \] and \[ g_1^{-1}g_2g_3^{-1}g_4=(x^3r)(s)(sr)=x^3r^2=x^5=x, \] as desired. QED.

Remark. In fact, we have \(T(G)< A(G)=Z(G)\) in this example.

References

[Cent] Central product of D8 and Z4, Groupprops, The Group Properties Wiki, https://groupprops.subwiki.org/wiki/Central_product_of_D8_and_Z4. Accessed: 3-25-26.

[LT25] L. Ta. Good involutions of conjugation subquandles (2025). Preprint, arXiv:2505.08090.

[YT23] Y. Taniguchi. Good involutions of generalized Alexander quandles, J. Knot Theory Ramifications 32 (2023), no. 12, Paper No. 2350081, 7. MR4688855