A NOTE ON LOWER NEAR FRATTINI SUBGROUPS

Kandasamy Muthuvel

University of Wisconsin-Oshkosh

Theorem 1 in [1] reads, “Let H be a normal subgroup of a group G such that the order of H is prime. Let λ(G) denote the set of all non–near generators of G. Then λ(G) ∩ H = {1} if and only if G nearly splits over H.” The purpose of this note is to show that Theorem 1 in [1] may be improved as follows: If H is a normal subgroup of a group G and H is of prime order, more generally, if H is a finite cyclic normal subgroup of a group G, then H ⊆ λ(G) and G does not nearly split over H. We also prove that if the condition “H is finite” is replaced by “H is infinite” in the above statement, then λ(G) ∩ H = {1} if and only if G nearly splits over H.

We first recall some definitions (see [1] or [2]).

\underline{Definition 1}. An element g of a group G is a non–near generator of G if S ⊆ G and |G :< g, S > | is finite implies |G :< S > | is finite. The set of all non–near generators of G, denoted by λ(G), is called the lower near Frattini subgroup of G.

\underline{Definition 2}. Let H be a normal subgroup of a group G. We say that G nearly splits over H if there exists a subgroup K of G such that |G : K| is infinite, |G : HK| is finite and

⋂_{g∈G} g^{−1}(H ∩ K)g = {1}.

\underline{Lemma 1}. If S is a subset of a group G and x is an element of G such that |G :< S > | is infinite and < x > is a finite normal subgroup of G, then |G :< h, S > | is infinite for every h ∈< x >.

\underline{Proof}. Let g ∈ G and |x| = n. Then

g < x, S >= g < x >< S >= ⋃_{1≤i≤n} gx^{i} < S > .

That is, any left coset of < x, S > in G is a finite union of left cosets of < S > in G and hence if |G :< x, S > | is finite, then G is a finite union of left cosets of < S > in G (i.e.

|G :< S > | is finite). Thus |G :< x, S > | is infinite. Now let h ∈< x >. Then h = x^{m} for some positive integer m. Since < x, S >⊇< x^{m}, S > and |G :< x, S > | is infinite, |G :< x^{m}, S > | is infinite. This completes the proof.

\underline{Proposition 1}. If H is a finite cyclic normal subgroup of a group G, then H ⊆ λ(G) and G does not nearly split over H.

\underline{Proof}. If H ⊈ λ(G), then there exists an h in H such that h is not in λ(G). h ∉ λ(G) implies that there exists a subset S of G such that |G :< S > | is infinite and |G :< h, S > | is finite. This is impossible by Lemma 1. Let H =< x >. If G nearly splits over H, then there exists a subgroup K of G such that |G : HK| = |G :< x, K > | is finite and |G : K| is infinite. Again by Lemma 1, this is impossible.

\underline{Remark 1}. It is interesting to note that the proof of part 1 of Proposition 1 works for any subgroup H of G such that every cyclic subgroup of H is a finite normal subgroup of G.

\underline{Lemma 2}. Let < x > be an infinite cyclic normal subgroup of a group G. If S is a subset of G such that |G :< S > | is infinite and |G :< x, S > | is finite, then < x > ∩ < S >= {1}.

\underline{Proof}. Suppose < x > ∩ < S >≠ \{1\}. Then there exists a smallest positive integer n such that x^{n} ∈< x > ∩ < S >. Hence

< x >< S >= ⋃_{1≤i≤n} x^{i} < S > .

This together with the hypothesis |G :< x, S > | is finite imply that G is a finite union of left cosets of < S > in G, which contradicts that |G :< S > | is infinite. Thus < x > ∩ < S >= {1}.

\underline{Theorem 1}. Let H be an infinite cyclic normal subgroup of a group G. Then λ(G) ∩ H = {1} if and only if G nearly splits over H.

\underline{Proof}. Let H =< x >. If λ(G) ∩ H = \{1\}, then there exists a subset S of G such that |G :< S > | is infinite and |G :< x, S > | is finite. By Lemma 2 and by using the fact that < x, S >=< x >< S >, it can be easily seen that G nearly splits over H. Conversely, if G nearly splits over H, then there exists a subgroup K of G such that |G : K| is infinite and |G : HK| is finite. |G : HK| = |G :< x, K > | is finite implies that for i ≠ 0, |G :< x^{i}, K > | is finite, because

< x, K >=< x > K ⊆ ⋃_{−|i|≤j≤|i|} x^{j} < x^{i}, K > .

Hence for i ≠ 0, x^{i} ∉ λ(G). Thus λ(G) ∩ H = {1}.

\underline{Remark 2}. Theorem 1 in [1] reads: “Let H be a normal subgroup of a group G such that the order of H is prime. Then λ(G) ∩ H = {1} if and only if G nearly splits over H. In other words λ(G) ∩ H ≠ {1} if and only if G does not nearly split over H.” The proof of Theorem 1 in [1] is correct, but it follows from Proposition 1 that under the hypotheses of Theorem 1 in [1], we get a much stronger conclusion, namely H ⊆ λ(G) and G does not nearly split over H.

\underline{References}

1. M. K. Azarian, “On Lower Near Frattini Subgroups and Nearly Splitting Groups,” Missouri J. Math. Sci. 2 (1990), 18-25.

2. J. B. Riles, “The Near Frattini Subgroups of Infinite Groups, ” J. of Algebra 12(1969), 155-171.