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A061350 Maximal size of Aut(G) where G is a finite Abelian group of order n. 4


%S 1,1,2,6,4,2,6,168,48,4,10,12,12,6,8,20160,16,48,18,24,12,10,22,336,

%T 480,12,11232,36,28,8,30,9999360,20,16,24,288,36,18,24,672,40,12,42,

%U 60,192,22,46,40320,2016,480,32,72,52,11232,40,1008,36,28,58,48,60,30,288

%N Maximal size of Aut(G) where G is a finite Abelian group of order n.

%C a(n) is multiplicative; if n = p^m is a prime power the maximal size of Aut(G) is attained by the elementary Abelian group G =(C_p)^m and then Aut(G) is GL(m,p) and a(n) = (p^m - 1)*(p^m - p)*...*(p^m - p^(m-1)). For general n the maximum will be for the direct product of the (C_p)^m over the prime powers dividing n and then the automorphism group is the direct product of the GL(m,p).

%C Equivalently, maximal size of Aut(G) where G is a nilpotent group of order n. - _Eric M. Schmidt_, Feb 27 2013

%H T. D. Noe, <a href="/A061350/b061350.txt">Table of n, a(n) for n=1..1024</a>

%p A061350 := proc(n) local ans, i, j; ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*(mul(ifactors(n)[2][i][1]^ifactors(n)[2][i][2] - ifactors(n)[2][i][1]^(j - 1), j = 1..ifactors(n)[2][i][2])): od: RETURN(ans) end:

%t a[p_?PrimeQ] := p-1; a[1] = 1; a[n_] := Times @@ (Product[#[[1]]^#[[2]] - #[[1]]^k, {k, 0, #[[2]]-1}]& /@ FactorInteger[n]); Table[a[n], {n, 1, 63}] (* _Jean-Fran├žois Alcover_, May 21 2012, after Maple *)

%Y Cf. A059773, A002884, A053290, A053292, A053293.

%K nonn,mult,nice,easy

%O 1,3

%A Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 07 2001

%E More terms from _Vladeta Jovovic_, Jun 12 2001

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Last modified December 20 21:34 EST 2014. Contains 252289 sequences.