\documentstyle[12pt]{amsart} \textheight=574pt \textwidth=432pt \oddsidemargin=18pt \evensidemargin=18pt \topmargin=14pt %\usepackage{amstex} %\usepackage{xypic} %\usepackage{amscd} %\usepackage{fancyheadings} %\pagestyle{fancy} %\leftmargin=0pt %\rightmargin=0pt %\tolerance=1000 %\voffset=-2truecm %\hoffset=-1.5truecm \parindent=0pt \renewcommand\baselinestretch{1.1} \def\F{\hfill $\Box$} \def\hang{\hangindent\parindent} \def\textindent#1{\indent\llap{#1\enspace}\ignorespaces} \def\re{\par\hang\textindent} \def\gz{\ifmmode{Z\hskip -4.8pt Z} \else{\hbox{$Z\hskip -4.8pt Z$}}\fi} \newtheorem{thm}{Theorem}[section] \newtheorem{prop}[thm]{Proposition} \newtheorem{lemma}[thm]{Lemma} \newtheorem{rmk}[thm]{Remark} \newtheorem{defn}[thm]{Definition} \newtheorem{cor}[thm]{Corollary} \newtheorem{exam}[thm]{Example} \newtheorem{notn}[thm]{Notation} \newtheorem{notdef}[thm]{Notation and Definitions} \newtheorem{DT}[thm]{Definition and Theorem} \title[Certain Applications of the Burnside rings]{Certain applications of the Burnside rings and ghost rings in the representation theory of\\ finite groups (II)} \date{} \author{K. K. Nwabueze} \address{K. K. Nwabueze\\ Mathematical Sciences Research Institute\\ 1000 Centennial Drive\\ Berkeley, CA 94720-5070, U.S.A.} \email{nwabueze@@msri.org} \author{F. Van Oystaeyen} \address{ F. Van Oystaeyen\\ Department of Maths and Computer Science\\ University of Ant\-wer\-pen (UIA)\\ 2610 Wilrijk, BELGIUM} \email{voyst@@uia.ua.ac.be} \thanks{Dr. Nwabueze's Research at MSRI is supported in part by the University of Antwerpen (UIA) and the United States National Science Foundation grant DMS-9022140.} \begin{document} %\lhead{\bfseries \thepage} %\chead{} %\rhead[\fancyplain{}{\bfseries\rightmark}]{\fancyplain{}{\bfseries\leftmark}} %\lfoot{} %\cfoot{} %\rfoot{} %\setlength{\headrulewidth}{1pt} %\setlength{\footrulewidth}{1pt} \maketitle \begin{abstract} Using the Burnside ring theoretic methods a new setting and a complete description of the Artin exponent $A(G)$ of finite $p$-groups was obtained in \cite{Nw}. In this paper, we compute $A(G)$ for any finite group $G$ -- hence providing the global version of \cite{Nw}. \end{abstract} \section{Introduction} This is a continuation of the paper \cite{Nw}, where we obtained a new setting for the Artin exponent of a finite group and described these exponents for finite $p$-groups. In this note we show that the restriction made in \cite{Nw} that $G$ is a $p$-group can be remove. In otherwords the Artin exponent for any finite group $G$ depends on the value of a $p$-subgroup of $G.$ We claim that the Artin exponent $A(G\ ,\ {\cal U})$ of a finite $G$ is equal to $1$ if and only if $G$ is cyclic whereas $A(G\ ,\ {\cal U})$ difers by $p$ from the order of a $p$-subgroup of $G$ or is equal to the order of a $p$-subgroup of $G$ if $G$ is noncylic and theorem 4.2 dictates which. If the featuring $p$-subgroups of $G$ is Dihedral, Quoternion or semi-dihedral, then $A(G\ ,\ {\cal U})$ is $2$ or $4$ and theorem 4.4 dictates which. In section 3 we show that we can reduce our problem essentially to the case where $G$ is a finite $p$-groups. For the convenience of the reader, we recall in section 2 some results in Burnside ring theory and the useful results in \cite{Nw}. Finally in section 4, we described the Artin exponent for finite group $G.$ \medskip {\bf Some notation:} $G$ in this paper denotes a finite group. Let ${1_{G}}$ denote the trivial subgroup of $G;$ $p,$ a prime number; $$, subgroup of $G$ generated by $T;\ $ $_{N},$ the normal subgroup of $G$ generated by $T;\ $ $g_{h},$ the $g$ conjugate of $h;\ $ $[g\ ,\ h],$ the commutator of $g$ and $h\ -\ $ in particular $ [S\ ,\ T] := < [s\ ,\ t]\ |\ s \in S,\ t \in T>. $ Let $G^{sub/\sim}$ denote a set of representatives of the conjugacy classes of subgroups of $G$ and $Z(G)$ denote the center of $G.$ Let $G_{s} := \{ g \in G\ |\ gs = s\ \forall\ s \in S \},$ the stabilizer subgroup of $s.$ \section{Preliminaries} Let $G$ be a finite group. The set of isomorphism classes of finite left $G$-sets form a commutative semi ring $B^{+}(G)$ with respect to disjoint union and cartesian product. Using the Grothendieck ring construction on $B^{+}(G)$ one obtains the corresponding Grothendieck ring $B(G)$ associated with $B^{+}(G).$ This resulting ring is called the Burnside ring $B(G)$ of $G.$ So the Burnside ring is the Grothendieck ring of finite $G$-sets which is generated as an algebra over $Z$ by the isomorphism classes of finite left $G$-sets $S_{1}\ S_{2}\ .\ .\ .\ $ subject to the relations $ S_{1} - S_{2} = 0 \ \mbox{ if } S_{1} \cong S_{2}; \ \ $ $ S_{1} + S_{2} - (S_{1} \sqcup Y) = 0 \ \ $ and $ S_{1} \cdot S_{2} - (S_{1} \times S_{2}) = 0. $ In otherwords, the elements of the Burnside ring are the virtual $G$-sets, that is the formal difference $ S_{1} - S_{2} $ of isomorphism classes of finite $G$-sets $S_{1},\ S_{2}.$ Furthermore one has that $ S_{1} - S_{2}\ =\ S'_{1} - S'_{2}\ \Longleftrightarrow\ S_{3} \sqcup S_{1} \sqcup S'_{2}\ \cong\ S_{3} \sqcup S_{2} \sqcup S'_{1} $ where $S'_{1},\ S'_{2} $ and $S_{3}$ are $G$-sets. This of course is equivalent with $S_{1} \sqcup S'_{2}\ \cong\ S_{2} \sqcup S'_{1}.$ For any subgroup $ U \leq G$ of $G$ the map $\chi_{U} :\ S\ \longrightarrow\ \#\{ s \in S\ \ |\ \ U \leq G_{s} \},$ from $S$ onto the number of $U$ invariant elements in $S,$ induces a homomorphism from $ B(G)$ into $Z$ -- the integers. Using standard arguments concerning the Burnside ring the following statements are easily verified (see \cite{Dr}). \medskip (a) All homomorphism from $B(G)$ into $Z$ is of the form $\chi_{U}$ defined above. \medskip (b) For any subgroup $U,\ V \leq G$ one has that $\chi_{U}\ =\ \chi_{V}$ if and only if $ U \sim V$ ($U$ is $G$-conjugate to $V$). \medskip (c) The product map $ \chi := \mathop{\Pi}\limits_{i=1}^{k} \chi_{i} : B(G) \longrightarrow \mathop{\Pi}\limits_{i=1}^{k} Z$ of $k$ different homomorphisms is injective and maps $B(G)$ onto a subring of finite index $\mathop{\Pi}\limits_{i=1}^{k}(N_{G}(U_{i}\ :\ U_{i})$ of $\mathop{\Pi}\limits_{i=1}^{k} Z$ -- this way identifying $\mathop{\Pi}\limits_{i=1}^{k} Z$ with the integral closure $\tilde{B}(G)$ of $B(G)$ in its total quotient ring $ Q \otimes_{z} B(G) \cong \mathop{\Pi}\limits_{i=1}^{k} Q$ (see \cite{Dr},\cite{Di}). The product $\mathop{\Pi}\limits_{i=1}^{k} Z$ is called the {\it Ghost ring} of $G.$ \medskip It is well known (see \cite{Dr}) that there are canonical set of congruence relation where by one can characterize the image $\chi(B(G))$ as a subgroup of $\mathop{\Pi}\limits_{i=1}^{k} Z.$ More exactly, if $ U \unlhd V \leq G $ then for a $G$-set $X$ the Burnside lemma applied with respect to the $V/U$-set $X^{U},$ implies that $ \mathop\sum\limits_{i=i}^{k} \chi_{<{\overline v}>} (X) \equiv 0 (mod\ (V:U)), $ where $<{\overline v}>$ denotes the subgroup of $V \leq G$ generated by the coset $ {\overline v} \subseteq V.$ \medskip \begin{lemma} Identifying $B(G)$ with its image in ${\tilde B}(G)$ with respect to the map $\mathop\Pi\limits_{U \in G^{sub/\sim}} \chi_{U}$ one has that $\{ n \in Z\ |\ n{\tilde B}(G) \subseteq B(G)\}\ =\ |G|Z.$ \end{lemma} Because for $U,\ V \leq G,$ one has $\chi_{U} = \chi_{V}$ if and only if $U$ and $V$ are $G$-conjugate and for $x,\ y \in B(G),$ one has $\chi_{U}(x) = \chi_{U}(y)$ for all $U \leq G$ if and only if $x = y$ it follows that we can identify each $x \in B(G)$ with the associated map (also denoted by $x$) $U \longrightarrow \chi_{U}(x),$ from the set of all subgroups of $G$ into $Z.$ Therefore we can consider $B(G)$ in a canonical way as a subring of the ghost ring $\tilde{B}(G).$ For more detail on the Burnside rings see \cite{Dr}. We recall the following simple facts (see any standard book on group theory). \begin{defn} Let $G$ be a $2$-group with generators $g$ and $h.$ We say that:\\{\bf(A)} $G$ is quaternion {\bf (Q)} if $g,$ $h$ have the relations $$ g^{2^{n}} = 1,\ \ h^{2} = g^{2^{n-1}},\ \ hgh^{-1} = g^{-1}.$$ {\bf (B)} $G$ is dihedral {\bf (D)} if $g,$ $h$ have the relations $$ g^{2^{n}} = 1,\ \ h^{2} = 1,\ \ hgh^{-1} = g^{-1}.$$ {\bf (C)} $G$ is semi-dihedral {\bf (SD)} if $g,$ $h$ have the relations $$ g^{2^{n}} = 1,\ \ h^{2} = 1,\ \ hgh^{-1} = g^{-1 + 2^{n-1}}.$$ \end{defn} \medskip \begin{lemma} Let $G$ be a noncyclic abelian $p$-group $(p \ne 2)$ and $U \unlhd G,$ $|U| = p.$ Then there exists a normal subgroup $ V \unlhd G $ of $G$ of order $p^{2},$ containing $U$ and isomorphic to $ C_{p} \times C_{p}. $ \end{lemma} \begin{lemma} Let $G$ be a finite $p$-group and\ $U$\ a subgroup of $G$ of order less than\ $p^{\beta}.$\ For $g \in G$ we have that $$ is cyclic of order $p^{\beta}$ if and only if $U \subseteq .$ \end{lemma} \begin{lemma} Let $G$ be a finite $p$-group and let $U$ and $H$ be subgroups of $G$ such that $U$ is a cyclic normal subgroup with $ U \subseteq H.$ Let ${\cal{C}}(H)$ denote the set of all cyclic subgroups $V$ of $H$ with $ U \leq V $ and $(V : U) = p,$ and let $ {\cal{C^{\prime}}}(H) = {\cal{C^{\prime}}}_{U}(H)$ denote the subset of $ {\cal{C}}(H),$ consisting of those $ V \in {\cal{C}}(H)$ which are normal in $H.$ We have:\\ (a.)\ $|{\cal{C}}(H)|\ \equiv\ |{\cal{C'}}(H)|\ \mbox{(mod } p).$\\ (b.)\ $ H^{'} = H^{'}_{U} := \{ h \in H\ |\ [h\ ,\ H] \subseteq U \}$ is a normal subgroup of $H.$\\ (c.)\ $ {\cal{C^{'}}}(H) = {\cal{C}}(H^{'})$\\ (d.) If $G^{'} \subseteq H \unlhd G$ then $|{\cal{C}}(H)|\ =\ |{\cal{C}}(G^{'})|(mod\ p). $ \end{lemma} \begin{lemma} With the above notation, one has that if $|U| = p \ne 2$ and $[G\ ,\ G] \subseteq U,$ then the following are eqivalent:\\ (i) $|{\cal{C}}(G)| = 1 $ \\ (ii) $ | {\cal{C}}(G) | \not\equiv 0(\mbox{mod }\ p) $\\ (iii) $ G $ is cyclic \end{lemma} \begin{lemma} If $G$ is abelian, then the following are equivalent:\\ (i) $|{\cal{C}}(G)| = 1 $ \\ (ii) $ | {\cal{C}}(G) | \not\equiv 0(\mbox{mod }\ p) $\\ (iii) $ G $ is cyclic \end{lemma} \begin{lemma} If $|U| = p \ne 2,$ then the following are eqivalent:\\ (i) $|{\cal{C'}}(G)| = 1 $ \\ (ii) $ | {\cal{C'}}(G) | \not\equiv 0(\mbox{mod }\ p) $\\ (iii) $ G' $ is cyclic \end{lemma} \begin{lemma} With the definitions and assumptions of (3.1), if $ G $ is a $ 2$-group, one has\\ (a.)\ $ |{\cal C}(H)| \equiv 1(2) \Rightarrow Z(G) \mbox { is cyclic. } $\\ (b.)\ If $[G\ ,\ G] \subseteq U$\ and\ $|{\cal C}(H)|$\ is odd, then $G$ is cyclic or nonabelian of order $8.$\\ (c.)\ If $ H \leq G $ with $ [H\ ,\ H] \subseteq U \subseteq H $ and $| \{ V \in {\cal C}(H)\ |\ V \subseteq H \}| \equiv 1 (\mbox{mod}\ 2),$ then $H$ is cyclic or nonabelian of order $8.$ \\ (d.)\ If $|{\cal C}(H)| \equiv 1 (\mbox{mod}\ 2),$ then either $ H'$ is cyclic or $G$ is nonabelian of order $8.$\\ (e.)\ If $ |{\cal C}(H)| \equiv 1(\mbox{mod}\ 2), $ then $G$ contains a cyclic normal subgroup of index $2.$\\ \end{lemma} \medskip In \cite{Nw}, we characterized the elements in $\tilde{B}(G)$ which are in $B(G)$ in the following way: \begin{DT} For ${\cal{U}}$ a family of cyclic subgroups of $G.$ Define $e_{{\cal{U}}} \in \tilde{B}(G)$ by $e_{{\cal{U}}}(U) = 1$ if $U$ is contained in ${\cal U}$ and $0$ otherwise. Then one has that $|G| \cdot e_{{\cal{U}}}(U) \in B(G).$ Furthermore define $A(G\ ,\ {\cal{U}}) := min(n \in N \mid n \cdot e_{{\cal{U}}} \in B(G)).$ The integer $A(G\ ,\ {\cal U})$ is said to be the {\it Artin exponent} of $G.$ One also has that $ A(G\ ,\ {\cal{U}}) $ divides $|G|$ (see \cite{Nw}). \end{DT} In \cite{Nw} we also gave a Burnside ring theoretic proof of the following well known results of Lam (see \cite{La}). \begin{thm} Let $G$ be a finite $p$-group. One has that $A(G\ ,\ {\cal U}) = 1$ if and only if $G$ is cyclic. \end{thm} \begin{thm} Let $G$ be a noncyclic finite $p$-group $(p \ne 2)$ of order $p^{\alpha}.$ Then one has $A(G\ ,\ {\cal U}) = p^{\alpha -1}. $ \end{thm} \begin{thm} If $p = 2 $ and $G$ is a noncyclic $2$-group of order $2^{\alpha}$ then $A(G) = 2^{\alpha -1},$ excluding the cases where $G$ is the Quaternion or Dihedral in which cases one has that $A(G\ ,\ {\cal U}) = 2.$ \end{thm} These results provides the complete computation of $A(G\ ,\ {\cal U})$ where $G$ is a $p$-group. \medskip \section{Reduction to $ p$-Groups} Let $p$ be a prime and consider $Z_{p},$ the localization of $ Z $ at $ p, $ that is\\ $Z_{p}\ :=\ \{ a/b\ \ |\ \ a \in Z,\ b \in Z - pZ \}\ \subseteq\ Q.$ Put $B_{p}(G)\ :=\ Z_{p}\ \otimes_{z}\ B(G),$ and $ \tilde{B}_{p}(G)\ :=\ Z_{p}\ \otimes_{z}\ \tilde{B}(G).$ One has that $ Z_{p} \cdot \chi ( B(G) )\ \cong\ B_{p}(G)\ :=\ Z_{p}\ \otimes_{z}\ B(G)$ coincides with the subgroup of ${\tilde B}_{p}(G)$ for which the relations $\mathop\sum\ n \cdot e_{\cal U}\ \equiv\ 0(mod\ (V_{p}\ :\ U)),$ with $V_{p}$ representing a subgroup of $G$ between $U$ and $V$ for which $V_{p}/U$ is the Sylow $ p$-subgroup of $V/U$ (see \cite{Dr}). Therefore $e_{\cal U}$ is characterized as a subgroup of $ \mathop\Pi\limits_{i=1}^{k} Z $ by those relations, which runs through all the prime divisors of $|G|.$ Hence we have; \begin{lemma} An element $n \cdot e_{{\cal{U}}} \in \tilde{B}(G)$ is contained in $B(G)$ if and only if for every $U \unlhd V \leq G$ with $(V : U)$ a prime power one has $ \mathop{\sum}\limits_{vU \in V/U}n \cdot e_{{\cal{U}}}()\ \equiv\ \\ 0 (\mbox{mod}\ (V : U)),$ where $e_{{\cal{U}}}() := \# \{ vU \in V/U\ |\ \ \in\ {\cal{U}} \}.$ \end{lemma} \begin{cor} For $ U \unlhd V \leq G,$ assume that $ (V : U) = p^{\alpha} $ for some prime $ p.$ Consider the decomposition $ U = U_{p} \times U_{p^{'}} $ where $|U_{p}| = p^{\beta}$ is a power of $ p $ and $ |U_{p^{'}}| $ is prime to $p.$ Let $ V_{p} $ denote the Sylow subgroup of $ V. $ Then one has\\ (a) $ (V : U) = p^{\alpha} $ implies that $ V = V_{p} \propto U_{p^{'}} $ and $ V = V_{p} \times U_{p^{'}}$ if and only if $ U \leq Z(V),$ the center of $V.$ (Here $ \propto, $ resp. $ \times $ denotes the semidirect product, resp. direct product).\\ (b) $ (V : U) = (V_{p} : U_{p}).$\\ (c) If\ $ c(U\ ,\ V) := \#\{ vU \in V/U\ \ |\ \ \mbox{ is cyclic } \} $ and \\ $ c(U_{p}\ ,\ V_{p}) := \#\{ vU_{p} \in V_{p}/U_{p}\ \ |\ \ \mbox { is cyclic } \} $ then $ c(U\ ,\ V)\ =\ c(U_{p}\ ,\ V_{p}) $ if $ V = V_{p} \times U_{p^{'}}, $ that is $ U \leq Z(V). $ \end{cor} \begin{rmk} Observe that for\ $ vU \in V/U $ the group $ $ is cyclic only if\\ $ v \in C_{V}(U)\ :=\ \{ w \in V\ |\ wu = uw \mbox { for all } u \in U \}.$ This implies that $(V : C_{V}(U))$ must always divide the number $"n"$ in order for $(V\ :\ U)$ to divide $n \cdot {\bf c}(U\ ,\ V).$ So we have the following strategy for computing the Artin exponent. For each $p$ subgroup $V$ of $G,$ consider the set of all cyclic normal subgroups $U$ of $V$ with $U$ contained in the centre of $V$ and compute the minimum of those positive integers $n$ for which $(V : U)$ divides $n \cdot c(U\ ,\ V).$ Finally, it is easy to see that if $U \leq Z(V),$ for a $p$-group $V$ such that $ U $ cyclic one has that, $ {\bf c}(U\ ,\ V) = {\bf c}(U/U^{p}\ ,\ V/U^{p}), $ where $ U^{p} $ denotes the unique subgroup of $U$ of index $p.$ So we can restrict ourselves to the case where $|U| = p.$ \end{rmk} We now summarize with the following, \begin{thm} Let $G$ be a finite group. Then $A(G\ ,\ {\cal U})\ =\ A(G_{p}\ ,\ {\cal U}),$ where $G_{p}$ is a $p$-subgroup of $G.$ \end{thm} \section{Global Discussion} For this discussion it suffices to consider the following cases; \medskip \underline{\bf Case 1:} $G$ is cyclic. \medskip The first thing to note is that the proof of Theorem(2.8) in \cite{Nw} does not depend on the fact that $G$ is a $p$-group. For the sake of explicitness we state the following global version of the theorem. That is we remove the condition that $G$ is a $p$-group. \begin{thm} Let $G$ be a finite group. For $ n \in N $ and a subgroup $U \leq G,$ let $x_{n} \in \tilde{B}(G)$ be defined by $$ x_{n} = \begin{cases} n,& \text{ if $ U $ is cyclic }\\ 0,& \text{ otherwise } \end{cases} $$ Then $A(G\ ,\ {\cal U}) = 1$ if and only if $G$ is cyclic. \end{thm} {\bf Proof:} We check by congruences. Assume $U \unlhd V \leq G$ such that $U$ cyclic of order $p$ and $U$ is contained in the center of $V$, where $|V| = p^{\alpha},$ ($p$ a prime). Then $ x \in B(G)$ implies $\mathop\sum\limits_{vU \in V/U}x() = n \cdot \#\{vU \in V/U\ |\ \mbox{ is cyclic}\}.$ To compute $A(G\ ,\ {\cal U}),$ we let $V = G,$ that is $|G| = p^{\alpha}.$ These assumption remains valid for the balance of this paper. Now if $n = 1,$ then one has that\\ $\#\{gU \in G/U\ |\ \mbox { is cyclic }\}\ \equiv \ 0(mod\ (G\ :\ U)).$ So\\ $(G\ :\ U)\ \ |\ \ \#\{gU \in G/U\ |\ \mbox{ is cyclic } \}.$ This implies\\ $|G|\ |\ \#\{ g \in G\ \ |\ \ \mbox { is cyclic } \}.$ So for all $g \in G,$ one has that $$ is cyclic. This implies that $G$ is cyclic since if $G$ is noncyclic then from (2.3) there exists a normal subgroup $ W\ \unlhd\ G$ such that $W$ contains $ U $ and $ W\ \cong\ C_{p} \times C_{p}\ \cong\ U \times C_{p}\ =\ U \times $ for some $g \in G.$ That is $ \ =\ W$ for some $g \in G.$ So $G$ is cyclic. \medskip The converse is obvious. \F \bigskip \underline{{\bf Case 2:}} $G$ is noncylic. \medskip \begin{thm} With the assumptions of Theorem 4.1 one has that if $G$ is noncyclic then $$ A(G\ ,\ {\cal U}) := \begin{cases} p^{\alpha},& \text{if $G'\ :=\ \{ g \in G\ |\ [g\ ,\ U] \subseteq U \}$ is cyclic}\\ p^{\alpha - 1},& \text{ if otherwise}. \end{cases} $$ \end{thm} {\bf Proof:} Since $G$ is noncyclic, then one obviously has that $|G|$ does not divide $\#\{g \in G\ |\ \mbox{ is cyclic} \}.$ If $G$ is abelian recall from lemma 2.3 that there exist a normal subgroup $W$ of order $p^{2}$ of the form\\ $\{ W \unlhd G\ |\ U\ \leq\ W \cong C_{p} \times C_{p}\}$ and indeed we have (see lemma 2.5 and 2.7) that\\ ${\cal C}(W)\ :=\# \{C\ \leq\ W\ |\ C \mbox{ is cyclic}\ C \supseteq U \} \equiv 0(mod\ p).$ This implies $A(G\ ,\ {\cal U}) = p^{\alpha -1}.$ \medskip If $G$ be nonabelian and $G'$ is noncyclic then repeating the above argument (by setting $G = G',$ one has that $A(G\ ,\ {\cal U}) = p^{\alpha - 1}.$ If $G'$ is cyclic then it is easy to derive (from 2.8) that $\#\{g \in G\ |\ \mbox{ is cyclic }\}\ \not\equiv\ 0 (mod\ p),$ in which case $A(G\ ,\ {\cal U})\ =\ p^{\alpha}.$ \F \medskip We now compute $A(G\ ,\ {\cal U})$ for the special groups $Q,\ D,\ $ and $SD.$ \begin{cor} Keeping the assumptions of (4.1) one has that if $G$ is any of $Q,\ D,\ $ or $SD$ then $$ A(G\ ,\ {\cal U}) := \begin{cases} 4,& \text{if $G'$ is cyclic}\\ 2,& \text{ if otherwise}. \end{cases} $$ \end{cor} {\bf Proof:} Using the notation in section 3 one has here that if ${\cal C}(G) \equiv 0(mod\ 2)$ then by a repetition of the above arguments one has that $A(G\ ,\ {\cal U}) = 2^{\alpha - 1}.$ If ${\cal C}(G) \not\equiv 0(mod\ 2)$ then, from lemma 2.9, $G$ is cyclic or a nonabelian group of order $8.$ But by our assumption $G$ is noncyclic and so $G$ must be a nonabelian group of order $8.$ It is now easy to see (by simple computation using lemma 2.9 repeatedly) that $A(G) = 4$ when $G'$ is cyclic and $A(G) = 2$ otherwise. \F \pagestyle{plain} \begin{thebibliography}{} \bibitem{Dr1} A. 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