<html><head><title>[AutPGrp] 1 Introduction</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <h1>1 Introduction</h1><p> <p> Given an arbitrary finite group, the computation of its automorphism group is a very difficult task. Pioneer work in this area was carried out by Felsch & Neubüser (1970), whose algorithm used the output of their subgroup lattice program. A technique developed by Neubüser in the early 1970s sought to compute the automorphism group viewed as a permutation group acting on unions of certain conjugacy classes of the group. A similar method was implemented by Hulpke (1997) in the <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 library. Recently, Cannon & Holt (1999) presented a new algorithm which uses a ``hybrid group'' approach. <p> More efficient approaches are available to determine the automorphism group for groups satisfying certain properties. Following the work of Shoda (1928), Hulpke in 1997 implemented a practical method for finite abelian groups in the <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 library. Wursthorn (1993) adapted modular group algebra techniques to compute the automorphism groups of <var>p</var>-groups; the <font face="Gill Sans,Helvetica,Arial">GAP</font> 3 share package <font face="Gill Sans,Helvetica,Arial">Sisyphos</font> includes an implementation. Smith (1994) introduced an algorithm for finite solvable groups which is available in the <font face="Gill Sans,Helvetica,Arial">AutAg</font> share package of <font face="Gill Sans,Helvetica,Arial">GAP</font> 3. <p> Moreover, the <var>p</var>-group generation method of Newman (1977) and O'Brien (1990) can be modified to compute the automorphism group of a finite <var>p</var>-group as outlined in O'Brien (1995). This algorithm is implemented in the ANU <code>pq</code> C program. <p> Here we introduce a new function to compute the automorphism group of a finite <var>p</var>-group. The underlying algorithm is a refinement of the methods described in O'Brien (1995). In particular, this implementation is more efficient in both time and space requirements and hence has a wider range of applications than the ANU <code>pq</code> method. Our package is written in <font face="Gill Sans,Helvetica,Arial">GAP</font> code and it makes use of a number of methods from the <font face="Gill Sans,Helvetica,Arial">GAP</font> library such as the MeatAxe for matrix groups and permutation group functions. <p> The <font face="Gill Sans,Helvetica,Arial">GAP</font> 4 package <font face="Gill Sans,Helvetica,Arial">ANUPQ</font>, which is an interface to most of the functionality of the ANU <code>pq</code> C program, uses the <font face="Gill Sans,Helvetica,Arial">AutPGrp</font> package to compute automorphism groups of <var>p</var>-groups. <p> We have compared our method to the others available in <font face="Gill Sans,Helvetica,Arial">GAP</font>. Our package usually out-performs all but the method designed for finite abelian groups. We note that our method uses the small groups library in certain cases and hence our algorithm is more effective if the small groups library is installed. <p> A <font face="Gill Sans,Helvetica,Arial">GAP</font> 3 version of the methods implemented in this package is available via <a href="http://www-public.tu-bs.de:8080/~beick/so.html">http://www-public.tu-bs.de:8080/~beick/so.html</a> <p> <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>AutPGrp manual<br>August 2003 </address></body></html>
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