[GAP Forum] Minimal Faithful Permutation Representation Degree

[Saad]

Dear forum,

sorry, I may be confused by "reverse inequality"  which proved clearly in :

Minimal Permutation Representations of Finite Groups
Author(s): D. L. Johnson
Source: American Journal of Mathematics, Vol. 93, No. 4 (Oct. 1971), pp. 857-866

to be :

$\mu(G\times H)\leq \mu G +\mu H $  ( for any finite groups G, H)

$\mu(G\times H)\geq \mu G +\mu H $   ( for groups G, H  of coprime orders)

hence, $\mu(G\times H)= \mu G +\mu H $  ( for any finite group of coprime orders)


and equality is the ultimate value for any finite groups,





Saad Owaid.

On Sunday, September 1, 2019, 11:15:11 PM GMT+3, Saad <saadhala10 at hotmail.com> wrote:


Dear all,

Marston Conder refered to the paper :
MR0390040 (52 #10866)
Wright, D.<https://mathscinet-ams-org.ezproxy.auckland.ac.nz/mathscinet/search/author.html?mrauthid=225357>
Degrees of minimal embeddings for some direct products.
Amer. J. Math.<https://mathscinet-ams-org.ezproxy.auckland.ac.nz/mathscinet/search/journaldoc.html?id=776> 97 <https://mathscinet-ams-org.ezproxy.auckland.ac.nz/mathscinet/search/publications.html?pg1=ISSI&s1=229858> (1975), <https://mathscinet-ams-org.ezproxy.auckland.ac.nz/mathscinet/search/publications.html?pg1=ISSI&s1=229858> no. 4,<https://mathscinet-ams-org.ezproxy.auckland.ac.nz/mathscinet/search/publications.html?pg1=ISSI&s1=229858> 897–903.
20C99<https://mathscinet-ams-org.ezproxy.auckland.ac.nz/mathscinet/search/mscdoc.html?code=20C99>

Which states that the reverse of the inequality holds when the groups G, H have coprime orders

Alexander Hulpke pointed that it was a bug....its worth to mention that it may not be this case in general considering the orders of the groups, for example:

gap> A:=AlternatingGroup(5);; oA:=Order(A);
60
gap> S:=OneSmallGroup(17,IsSolvable,true);oS:=Order(S);
<pc group of size 17 with 1 generators>
17
gap> D:=DirectProduct(S,A);;
gap> Gcd(oS,oA);
1
gap> muA:=MinimalFaithfulPermutationDegree(A);
5
gap> muS:=MinimalFaithfulPermutationDegree(S);
17
gap> muD:=MinimalFaithfulPermutationDegree(D);
85




Saad Owaid



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