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{\bf On ${\rm STD}_6[18,3]$'s and ${\rm STD}_7[21,3]$'s Admitting a Semiregular Automorphism Group of Order 9}
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{\bf Kenzi Akiyama, Masayuki Ogawa and Chihiro Suetake}
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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.



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