Maximal order of NG-Transformation group

: In this paper, we consider the problem that the maximal order consider the groups that consisting of transformations we called NG-Transformation on a nonempty set A has no bijection as its element. We find the order of these groups not greater that ( n -1)!. In addition, we will prove our result by showing that any kind of NG-group in the theorem be isomorphic to a permutation group on a quotient set of A with respect to an equivalence relation on A .


Theorem 1.2 let  be an SHP-class, and suppose G=UV, where U and V are subnormal in G. then G = U V .
We can take the SHP-class to the class of p-groups, the class of nilpotent groups, etc.
theorem1.2 will imply Lemma 9.15, problem 9B.5, Corollary 9.27, problem 9C.2, as corollaries. Remark 1.3. It was noted in Sec.4 of [8] that if we replace the condition that  is an SHP-class by some weaker condition that the class  is such that whose composition factors all lie in some given set of simple groups then theorem 1.2 will fail in this case. Definition 1.2. let  be an SHP-class and G be a finite group. We denoted the maximal normal -subgroup of G by G.
We will consider the question that if G=UV with U,V subnormal in G then it holds that G =UV or not. If p is a prime and take the SHP-class  to be the class of all finite p-group, then for any finite group G will be Op(G) and we have the following theorem, which is inspired by P.Iin, [12].   Preliminary: In this section, we review some basics concepts of the finite group theory that are assumed in our paper. The detailed here in lots of abstract algebra and finite group theory books for more including all treating of some of this material for example, in [11], [6],and, [9] would be good supplementary sources for the theory needed here. Theorem is revised version of Theorem 2, [14].

Theorem 2.5. Let f be an element in A A and ˆf be the induced transformation of f on A/~f,
i.e ˆf: Then the following hold: The exists a groups G A A containing f as the identity element iff f 2 =f.

(ii)
There is a groups G A A containing f as the identity element iff ˆf is abijective on A/~f.
The following two corollaries are from [14] and we make some corrections to the original proofs. Actually, we adopt the restriction of finiteness on A in the first corollary from the original one. And we used the finiteness on A in the second corollary; the original one did not use it.
It follows that f 2 = f as required.

Corollary 2.7. Suppose that A is a finite set and f is an element in A A . Then there is a group
G A A containing f as an element iff Im(f) = Im(f 2 ).
Proof. On one hand, suppose that there is a group G A A containing f as an element.
Let e be the identity element of G.  , aA, and therefore f = g. We conclude that  is injective. As a consequence,  is an isomorphism.

Definition 2.3. A subgroup H of a group G is called characteristic in G, denoted H
char G, if every automorphism of G maps H to itself, that is(H) = H for all Aut(G).

Remark 2.3. If H is characteristic in G in K and K is characteristic in G, then H is characteristic in G.
Let G be a finite group. We have the following two lemmas. They are from Section 2 of [8].

(c) G  is characteristic in G. (d) O (G) is characteristic in G.
The following lemma is a generalization of Problem 2A.1 in [7]. is the largest normal -subgroup of G. Suppose r > 1 and the containment holds for r-1. Let A1 = A⊲…⊲ Hr-1⊲Hr = G be a subnormal series from A to G: Then A O(G) by indctive hypothesis. Since O(G) char Hr-1 and Hr-1 ⊲G; O(G)⊲ G and then O(G) (Hr-1) O(G). We conclude that A O(G). In general, for any two subnormal -subgroups A and B, A,B O(G) and thus A,B  O(G) as wanted.

Proofs of Main Results
Now let A be a set having n letters written as {1, 2,…, n}. We have the following theorem, which is Theorem 1.1. Theorem 3.1. Let A be a set with cardinality n with n 3. Suppose NG is a group consisting of non-bijective transformations on A, where the binary operation on NG is the composition of transformations. Then the order of NG is not greater than (n-1)! and there are such groups having order (n-1)!: Proof. Let NG be a group consisting of non-bijective transformations on A. By Remark 2.1, we know that ~f=~g for any element f,g NG and we denote the common equivalence relation by ~. Note that NG is a group consisting of non-bijective transformations, then we see that the equivalence relation is not the equality relation =on A. Thus, we have that the quotient set A/~ has order less than n-1. Additionally, NG is isomorphic to a permutation group on A/~ by Theorem 2.8. It follows that the order of NG is less than (n-1)! as any permutation group on A/~ has order less than (n-1)!.
Proof. We use induction of the subnormal depth of U in G to prove the result. First, if the subnormal depth of U in G is one, i.e. U⊲G. Since U  is characteristic in U and U is normal in G we see that U  is normal in G.
LetG= G/U  . By the hypothesis,G=UV where U=U/ U  , V=V U  /U  . Thus,U is a normal-group ofG andV is subnormal inG. By Lemma 2.10, we have G  =V  . By Lemma 2.9 (b), G  = G  ,V  =V  =U  V  . By correspondence theorem, we have G  = U  V  ; as required. Now suppose that the subnormal depth of U in G is r with r > 1: Let U1 = U⊲…⊲ Ur⊲ G be a subnormal series from U to G with length r. By Dedekind's lemma, Ur =U(V Ur). As both U and V Ur are subnormal in Ur and U has subnormal depth r-1 in Ur, we obtain that (Ur)  = U  (V Ur)  by inductive hypothesis. Also, G = UrV with Ur normal in G and V subnormal in G, and hence G  = (Ur)  V  by the first paragraph of the proof. It  Proof. Since N is normal in G and the quotient group G=N/H is abelian, we deduce that the derived subgroup G' is contained in N. It follows that both U and V contain G' as a subgroup, which implies that U and V are normal in G. Obviously, G = UV. Assertion (i) holds. Note that the Sylow p-subgroup of G is not normal since x act on N faithfully and hence Op(G) has order less than p 2 . However, as y act on N trivially, N normalizes y which yields that y is a normal p-subgroup of G. It is easy to see that Op(G) = y. Both x and xy act faithfully on N, which yields that Op(U) = Op(V) = 1; as wanted.