Subnormality is permuting join-closed

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This article gives the statement, and possibly proof, of a subgroup property (i.e., subnormal subgroup) satisfying a subgroup metaproperty (i.e., permuting join-closed subgroup property)
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Statement

Verbal statement

A join of two subnormal subgroups that permute is again a subnormal subgroup. Moreover, its subnormal depth is bounded by a function of the subnormal depths of the individual subnormal subgroups.

Statement with symbols

Suppose H and K are subnormal subgroups of a group G, and suppose further that HK = KH, i.e., the product of subgroups is again a subgroup. Then, HK is a subnormal subgroup of G. Further, if the subnormal depth of H is h and the subnormal depth of K is k, the subnormal depth of HK is bounded by:

h[k]^h = hk(k+1)(k+2) \dots (k + h - 1).

By symmetry, it is also bounded by the function k[h]^k. When h \le k, h[k]^h is the smaller of the two numbers, hence the more relevant bound.

Related facts

Applications

Examples

Subnormal depth h of H Subnormal depth k of K Upper bound on subnormal depth of HK given by this statement Best known bound on subnormal depth of HK Lowest possible upper bound based on explicit example
0 anything 0 0 0
1 k k k k -- take H trivial.
2 2 12 4 (see 2-subnormal implies join-transitively subnormal) 3 (see 2-subnormality is not finite-join-closed)
2 k 2k(k + 1) 2k (see 2-subnormal implies join-transitively subnormal)  ?
3 3 180 180  ?
3 4 360 360  ?

Facts used

  1. Modular property of groups: If A, B, C \le G such that A \le C we have A(B \cap C) = AB \cap C.
  2. Subnormality is normalizing join-closed: If A, B \le G are subnormal of depths a,b, and B \le N_G(A), then AB is a subnormal subgroup of depth at most ab.
  3. Subnormality satisfies intermediate subgroup condition: If H \le L \le G and H is h-subnormal in G, H is h-subnormal in L.
  4. Subnormality is finite-relative-intersection-closed: Suppose A is a-subnormal in C and B \le C such that B is b-subnormal in some subgroup D of C containing both A and B. Then, A \cap B is (a + b)-subnormal in C.

Proof

This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format

Given: A group G, a subnormal subgroup H of subnormal depth h in G, a subnormal subgroup K of subnormal depth k, such that HK = KH. Let J = HK.

To prove: J is a subnormal subgroup of G with subnormal depth at most hk(k+1) \dots (k + h - 1).

Proof: We break the proof into many steps. Note that we use permutability where we use the fact that the product HK is the subgroup generated by H and K.

The key idea is to construct one subnormal series, first going down the series, and then going back up the series using some bounds. Our first few steps are in the ambient group J, and our later steps shift focus to the ambient group G.

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation Commentary
1 H is h-subnormal in J Fact (3) H is h-subnormal in G -- Piece together. Shifting to ambient group J.
2 Define a subnormal series for H in J by H_0 = J, and H_{i+1} = H^{H_i}. Note that H_0 \ge H_1 \ge H_2 \ge \dots Still in ambient group J, viewing subnormality there.
3 H_h = H. Moreover, each H_i is i-subnormal in J. Steps (1), (2) This follows from the definition of subnormal depth. Still in ambient group J, viewing subnormality there.
4 H_i \cap K is (k + i)-subnormal in G Fact (4) K is k-subnormal in G Step (3) Piece together. In Fact (4), set A = K, B = H_i, C = G, D = J. Crucial step, where we switch to ambient group G.
5 H_i = H_{i+1}(K \cap H_i) Fact (1) H and K permute Step (2) [SHOW MORE] Preparing the ground for moving up from H to J via H_is in ambient group G.
6 K \cap H_i normalizes H_{i+1} Step (2) [SHOW MORE] Preparing the ground for moving up from H to J via H_is in ambient group G.
7 If H_{i+1} has subnormal depth s, then H_i is subnormal of depth at most s(k + i) Fact (2) -- Step (4) [SHOW MORE] Single step up the ladder from H to J via H_is, in ambient group G.
8 The final result H is h-subnormal in G) Steps (2), (7) [SHOW MORE] Combine multiplicative factors accumulated in each step of the climb up.

References

Textbook references

  • A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, More info
  • Subnormal subgroups of groups by John C. Lennox and Stewart E. Stonehewer, Oxford Mathematical Monographs, ISBN 019853552X, Page 5, Section 1.2 (First results on joins), Theorem 1.2.5, More info