[Turkmath:9315] MSGSU Seminer: Ergün Yaraneri / Abelian Groups With Isomorphic Intersection Graphs / 07.11.13

[Mustafa Topkara]

Mimar Sinan Güzel Sanatlar Üniversitesi - Matematik Bölümü
Genel Seminer

Konuşmacı:
Ergün Yaraneri (İstanbul Teknik Üniversitesi)

Baslik:
Abelian Groups With Isomorphic Intersection Graphs

Özet:
(PDF formatında ektedir)
Let $G$ be a group. The intersection graph $\mathcal{G} (G)$ of $G$ is an
undirected graph without loops and multiple edges defined as follows: the
vertex set is the set of all proper nontrivial subgroups of $G,$ and there
is an edge between two distinct vertices $X$ and $Y$ if and only if $X\cap
Y\neq 1$ where $1$ denotes the trivial subgroup of $G.$ It was conjectured
in ``[B. Zelinka, {\it Intersection graphs of finite abelian groups},
Czechoslovak Mathematical Journal, Volume: 25, Issue: 2, Pages: 171-174,
(1975)]" that two (non cyclic) finite abelian groups with isomorphic
intersection graphs are isomorphic. We study this conjecture and show that
it is almost true. For any finite abelian group $D$ let $D_{nc}$ be the
product of all noncyclic Sylow subgroups of $D.$ Our main result is that:
given any two (nontrivial) finite abelian groups $A$ and $B,$ their
intersection graphs $\mathcal{G} (A)$ and $\mathcal{G} (B)$ are isomorphic
if and only if the groups $A_{nc}$ and $B_{nc}$ are isomorphic, and there
is a bijection between the sets of (nontrivial) cyclic Sylow subgroups of
$A$ and $B$ satisfying a certain condition. So, in particular, two finite
abelian groups with isomorphic intersection graphs will be isomorphic
provided that one of the groups has no (nontrivial) cyclic Sylow subgroup.
Our methods are elementary.

Yer:
MSGSÜ Bomonti Kampüsü (Harita <http://math.msgsu.edu.tr/iletisim.html>),
Matematik Bölümü Seminer Odası.

Zaman:
7 Kasım 2013 Perşembe, 16:00
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