Normal fuzzy subgroups and fuzzy normal series of finite groups Fuzzy sets and systems 72 (1995) 379-383

FUZZY

sets and systems ELSEVIER

Fuzzy Sets and Systems 86 (1997) 121

Erratum

Normal fuzzy subgroups and fuzzy normal series of finite groups Fuzzy Sets and Systems 72 (1995) 379-383 M. A t i f A. M i s h r e f Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt Received March 1996

Abstract A counterexample of Theorem 3.1 in the paper by Ashref is presented which shows that the image of normal fuzzy subgroups may contain more than two elements. The conditions are given for this theorem to be satisfied.

Keywords." Fuzzy algebra; Fuzzy subgroups

1. Counterexample Example 1. Consider G the four group Z4 of residue classes and let A be a fuzzy subset defined as follows: A ( 0 ) = 1,

3 A ( l , 3 ) = ½, A ( 2 ) = ~.

It is clear that A is a fuzzy subgroup of G. Since G is abelian, i is normal. It is noted that Ira(A) = { 1, 3, ½}, which contradicts Theorem 3.1 in [1]. In the following we give a correction of this theorem.

Theorem 2. Let A be a normal fuzzy subgroup of a finite group G. Then Im(A) contains at most two elements if G/GJ is isomorphic to G/G~ for every t E Im(A). Proof. As in the second part of the proof of Theorem 3.1 [1]. Reference [1] M.A. Mishref, Normal fuzzy subgroups and fuzzy normal series of finite groups, Fuzzy Sets and Systems 72 (1995) 379-383.

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