Pith Number

pith:MRYUBYLE

pith:2026:MRYUBYLEMPPPHDR23FTHWN5RF7

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Greedy bases and relational complexity of diagonal type groups

Colva M. Roney-Dougal, Hong Yi Huang

Primitive groups of diagonal type satisfy Cameron's conjecture on greedy base sizes and have relational complexity at least 4 that is unbounded.

arxiv:2605.16032 v1 · 2026-05-15 · math.GR

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Claims

C1strongest claim

We determine the size of every base returned by the greedy algorithm when G is a primitive group of diagonal type, and hence prove Cameron's conjecture for these groups. We prove that if G is primitive of diagonal type then RC(G) ≥ 4, that this lower bound is attained by infinitely many such G, and that the relational complexity of the groups of diagonal type is unbounded.

C2weakest assumption

The analysis relies on the known structural description of primitive diagonal type groups (socle T^k acting on cosets of a diagonal subgroup) and on the fact that the greedy choice can be tracked via the orbits on the product space; if this structural description or the orbit calculations contain an undetected gap, the exact base sizes and the RC lower bound would not follow.

C3one line summary

For primitive diagonal type groups the greedy base sizes are computed exactly, proving Cameron's conjecture, while relational complexity is shown to be at least 4 with no upper bound.

References

31 extracted · 31 resolved · 0 Pith anchors

[1] M. Anagnostopoulou-Merkouri and T.C. Burness,On the regularity number of a finite group and other base-related invariants, J. Lond. Math. Soc.110(2024), Paper No. e70035, 65 pp 2024

[2] K.D. Blaha,Minimum bases for permutation groups: the greedy approximation, J. Algorithms13 (1992), no. 2, 297–306 1992

[3] J.N. Bray, D.F. Holt and C.M. Roney-Dougal,The maximal subgroups of the low-dimensional finite classical groups, Lond. Math. Soc. Lecture Note Ser., vol. 407, Cambridge Univ. Press, Cambridge, 2013 2013

[4] W. Bosma, J. Cannon and C. Playoust,TheMagmaalgebra system I: The user language, J. Symb. Comput.24(1997), 235–265 1997

[5] S. Brenner, C. del Valle and C.M. Roney-Dougal,Irredundant bases for soluble groups, Bull. Lond. Math. Soc.57(2025), 3013–3023 2025

Receipt and verification

First computed	2026-05-20T00:01:50.021688Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

647140e16463def38e3ad9667b37b12ff545c000a737157168230c77c64a1de6

Aliases

arxiv: 2605.16032 · arxiv_version: 2605.16032v1 · doi: 10.48550/arxiv.2605.16032 · pith_short_12: MRYUBYLEMPPP · pith_short_16: MRYUBYLEMPPPHDR2 · pith_short_8: MRYUBYLE

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/MRYUBYLEMPPPHDR23FTHWN5RF7 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 647140e16463def38e3ad9667b37b12ff545c000a737157168230c77c64a1de6

Canonical record JSON

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    "abstract_canon_sha256": "6aca950624da861a471db57371dbd9558c7d2038b426369f43b2805a1a93c32b",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.GR",
    "submitted_at": "2026-05-15T15:07:34Z",
    "title_canon_sha256": "a2e1dd1127b9fa043d5f863775129227f19537b338ac58f148dc8b371c05a81c"
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