On STD6[18, 3]'s and STD7[21, 3]'s Admitting a Semiregular Automorphism Group of Order 9
On STD6[18, 3]'s and STD7[21, 3]'s Admitting a Semiregular Automorphism Group of Order 9
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关于STD6[18, 3]和STD7[21, 3]承认9阶半正则自同构群
DOI：
10.37236/237
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
C. Suetake
中科院分区：
文献类型：
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作者：
Kenzi Akiyama;Masayuki Ogawa;C. Suetake
利用群环${\bfZ}[G]$刻划了对称断面设计${\rm std}_{\lambda}[k，u]$‘S在点和块上都有一个半正则自同构群$G$，其中包含一个阶为$u的自同构群.设$n_\lambda$是非同构的${\rm std}_{\lambda}[3\lambda，3]$‘S的个数，已知$n_1=1，\n_2=1，\n_3=4，n_4=1$，$n_5=0$。利用这一刻划，我们对点和块上都有一个9阶半正则非循环自同构群的S和S进行了分类。前者恰好产生20个非同构的$6[18，3]$‘S，后者恰好产生3个非同构的$7[21，3]$’S，它们得到$n_6\geq20$和$n_7\geq 5$，因为B.Brock和A.Murray在1991年构造了另外两个$\rm std}_7[21，3]$‘S。我们用了一台电脑进行研究。
We characterize symmetric transversal designs ${\rm STD}_{\lambda}[k,u]$'s which have a semiregular automorphism group $G$ on both points and blocks containing an elation group of order $u$ using the group ring ${\bf Z}[G]$. Let $n_\lambda$ be the number of nonisomorphic ${\rm STD}_{\lambda}[3\lambda,3]$'s. It is known that $n_1=1,\ n_2=1,\ n_3=4, n_4=1$, and $n_5=0$. We classify ${\rm STD}_6[18,3]$'s and ${\rm STD}_7[21,3]$'s which have a semiregular noncyclic automorphism group of order 9 on both points and blocks containing an elation of order 3 using this characterization. The former case yields exactly twenty nonisomorphic ${\rm STD}_6[18,3]$'s and the latter case yields exactly three nonisomorphic ${\rm STD}_7[21,3]$'s. These yield $n_6\geq20$ and $n_7\geq 5$, because B. Brock and A. Murray constructed two other ${\rm STD}_7[21,3]$'s in 1991. We used a computer for our research.
DOI：
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发表时间：
2008
期刊：
Discrete Mathematics 308
影响因子：
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作者：
K.;Watanabe;K. Watanabe;Y. Hiramine(with N.Ito);Y. Hiramine(with C. Suetake)
通讯作者：
Y. Hiramine(with C. Suetake)