In a previous paper of the first author a procedure was developed for counting the components of a graph through the knowledge of the components of one of its quotient graphs. Here we apply that procedure to the proper power graph of a finite group , finding a formula for the number of its components which is particularly illuminative when is a fusion controlled permutation group. We make use of the proper quotient power graph , the proper order graph and the proper type graph . All those graphs are quotient of . We emphasize the strong link between them determining number and typology of the components of the above graphs for . In particular, we prove that the power graph is -connected if and only if the type graph is -connected, if and only if the order graph is -connected, that is, if and only if either or none of is a prime.
DOI : 10.4171/RSMUP/138-3
Mots clés : Quotient graph, power graph, permutation groups
@article{RSMUP_2017__138__61_0, author = {Bubboloni, Daniela and Iranmanesh, Mohammad and Shaker, Seyed}, title = {Quotient graphs for power graphs}, journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova}, pages = {61--89}, publisher = {European Mathematical Society Publishing House}, address = {Zuerich, Switzerland}, volume = {138}, year = {2017}, doi = {10.4171/RSMUP/138-3}, mrnumber = {3743245}, zbl = {1387.05109}, url = {http://www.numdam.org/articles/10.4171/RSMUP/138-3/} }
TY - JOUR AU - Bubboloni, Daniela AU - Iranmanesh, Mohammad AU - Shaker, Seyed TI - Quotient graphs for power graphs JO - Rendiconti del Seminario Matematico della Università di Padova PY - 2017 SP - 61 EP - 89 VL - 138 PB - European Mathematical Society Publishing House PP - Zuerich, Switzerland UR - http://www.numdam.org/articles/10.4171/RSMUP/138-3/ DO - 10.4171/RSMUP/138-3 ID - RSMUP_2017__138__61_0 ER -
%0 Journal Article %A Bubboloni, Daniela %A Iranmanesh, Mohammad %A Shaker, Seyed %T Quotient graphs for power graphs %J Rendiconti del Seminario Matematico della Università di Padova %D 2017 %P 61-89 %V 138 %I European Mathematical Society Publishing House %C Zuerich, Switzerland %U http://www.numdam.org/articles/10.4171/RSMUP/138-3/ %R 10.4171/RSMUP/138-3 %F RSMUP_2017__138__61_0
Bubboloni, Daniela; Iranmanesh, Mohammad; Shaker, Seyed. Quotient graphs for power graphs. Rendiconti del Seminario Matematico della Università di Padova, Tome 138 (2017), pp. 61-89. doi : 10.4171/RSMUP/138-3. http://www.numdam.org/articles/10.4171/RSMUP/138-3/
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