Counting of the distinct fuzzy subgroups of the dihedral group D2p n
Küçük Resim Yok
Tarih
2009
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Maltepe Üniversitesi
Erişim Hakkı
CC0 1.0 Universal
info:eu-repo/semantics/openAccess
info:eu-repo/semantics/openAccess
Özet
In this paper, by using of an equivalence relation on fuzzy subgroup, we determine the number of distinct fuzzy subgroups of the dihedral group of order 2p n such that p is a prime and p ? 3. A fuzzy subset of a set X is mapping µ : X ? [0, 1]. Fuzzy subset µ of a group G is called a fuzzy subgroup of G if (G1) µ(xy) ? µ(x) ? µ(y)?x, y ? G; (G2) µ(x ?1 ) ? µ(x)?x ? G. The set of all fuzzy subgroup of a group G denoted by F (G). Let G be a group, and µ, ? ? F (G). Defined three equivalence relation as follow respectively: (i) We say that µ is equivalence ?, written as µ ? ? if Fµ = F? . (ii) We say that µ is equivalent to ?, written as µ ? ? if 1. µ(x) > µ(y) ? ?(x) > ?(y), for all x, y ? G. 2. µ(x) = 0 ? ?(x) = 0, for all x ? G. (iii) We say that µ is equivalence ?, written as µ 't ? if there exists an isomorphism f from suppµ to supp? such that for all x, y ? suppµ, µ(x) > µ(y) ? ?(f(x)) > ?(f(y)) Let G be a group and µ, ? ? F (G). We say that µ is equivalence ?, written as µ ?t ?, if and only if Fµ = F? and suppµ = supp?. The set of all fuzzy subgroups µ of G such that µ(e) = 1 denoted by F1(G) . The number of equivalence classes ? on F1(G) will be denoted by r ? G. Theorem. Suppose that p be a prime and p ? 3. If G is a dihedral group of order 2p n, then r ? G = nP?1 i=1 p i r ? (D2pn?i ) + pn?p p?1 + 2n+2 + 4p n ? 1.
Açıklama
Anahtar Kelimeler
Kaynak
International Conference of Mathematical Sciences
WoS Q Değeri
Scopus Q Değeri
Cilt
Sayı
Künye
Naraghi, H. (2009). Counting of the distinct fuzzy subgroups of the dihedral group D2p n. Maltepe Üniversitesi. s. 192.
