Natural numbers n which satisfy gnu(n)=n? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T16:12:11Z http://mathoverflow.net/feeds/question/46444 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46444/natural-numbers-n-which-satisfy-gnunn Natural numbers n which satisfy gnu(n)=n? Glen M Wilson 2010-11-18T01:37:12Z 2010-11-18T04:56:21Z <p>Are there any natural numbers \$n\$ (other than 1) for which \$gnu(n)=n\$? We define \$gnu(n)\$ to be the number of isomorphism classes of groups of order \$n\$. This question popped into my head today, and I couldn't come up with a proof one way or another. </p> <p>In the paper by Conway, et al. entitled <a href="http://www.math.auckland.ac.nz/~obrien/research/gnu.pdf%20%22Counting%20Groups%3A%20Gnus,%20Moas,%20and%20other%20Exotica%22" rel="nofollow">"Counting Groups: Gnus, Moas, and other Exotica</a>, Conjecture 10.1 implies that there should be no such natural number \$n\$ which satisfies \$gnu(n)=n\$. </p> <p>Does anybody have an argument to show that there is no natural number \$n\$ (other than 1) for which \$gnu(n)=n\$? If it is really straightforward, just say so and I'll work it out for myself when I find time. Thanks! </p>