Examples of non-$FSZ$ $p$-groups for primes greater than three

Keilberg, Marc

doi:10.1090/proc/13823

Cited by 3 publications

(28 citation statements)

References 17 publications

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“…Remark 3.4. The X p,j (p l ) play the same role that the Y p,j (p l ) played in [7]. The next few lemmas mirror the corresponding lemmas for the Y p,j (p l ).…”

Section: A Family Of F Sz Groups With Non-f Sz Central Quotientsmentioning

confidence: 65%

“…The group F (5, 1) is exactly SmallGroup(5 7 ,654) in GAP. This group was the basis for the above construction, in much the same fashion that S(5, 1) = SmallGroup(5 6 ,632) was the motivating example in [7].…”

Section: A Family Of F Sz Groups With Non-f Sz Central Quotientsmentioning

confidence: 96%

“…Several additional examples of F SZ + and non-F SZ groups can be found in [2,6,7,8,9,15]. It has been an open question as to whether or not there exists an F SZ group which is not F SZ + .…”

Section: Background and Notationmentioning

confidence: 99%

“…In this section we show how to obtain an infinite family of examples similar to the one from Theorem 2.7. For this, we will use the groups S(p, j) from [7]. It was shown in [7, Theorem 1.9] that S(p, j) was not F SZ p j for any prime p > 3 and j ∈ N. We will construct an F SZ p j group which has S(p, j) as one of its quotients by a central subgroup.…”

Section: A Family Of F Sz Groups With Non-f Sz Central Quotientsmentioning

confidence: 99%

“…Note for p > 2 that Q p,j = P p,j × Z p as defined in [7]; P p,j was not explicitly defined for p = 2 for technical reasons, but otherwise makes sense with P 2,1 = Z 4 . Let Q p,j have generators a 1 , ..., a p j , where a 1 has order p j+1 and the rest have order p.…”

Section: A Family Of F Sz Groups With Non-f Sz Central Quotientsmentioning

confidence: 99%

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Some behaviors of FSZ groups under central products, central quotients, and regular wreath products

Keilberg¹

2019

Journal of Algebra

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We show that any group G with a non-F SZm quotient by a central cyclic subgroup also provides a non-F SZm group of order m|G| obtained as a central product of G with a cyclic group. We then construct, for every prime p > 3 and j ∈ N, an F SZ p j group F such that there is a central cyclic subgroup A with F/A not F SZ p j . We apply these results to regular wreath products to construct an F SZ p-group which is not F SZ + for any prime p > 3. These give the first known examples of F SZ groups that are not F SZ + . We are also able to prove a few partial results concerning the F SZ properties for the Sylow subgroups of symmetric groups. In the appendix we enumerate all non-F SZ groups of order 5 7 .

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“…Remark 3.4. The X p,j (p l ) play the same role that the Y p,j (p l ) played in [7]. The next few lemmas mirror the corresponding lemmas for the Y p,j (p l ).…”

Section: A Family Of F Sz Groups With Non-f Sz Central Quotientsmentioning

confidence: 65%

Section: A Family Of F Sz Groups With Non-f Sz Central Quotientsmentioning

confidence: 96%

Section: Background and Notationmentioning

confidence: 99%

Section: A Family Of F Sz Groups With Non-f Sz Central Quotientsmentioning

confidence: 99%

Section: A Family Of F Sz Groups With Non-f Sz Central Quotientsmentioning

confidence: 99%

See 3 more Smart Citations

Some behaviors of FSZ groups under central products, central quotients, and regular wreath products

Keilberg¹

2019

Journal of Algebra

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The FSZ properties of sporadic simple groups

Keilberg¹

2019

J. Algebra Appl.

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We investigate a possible connection between the F SZ properties of a group and its Sylow subgroups. We show that the simple groups G 2 (5) and S 6 (5), as well as all sporadic simple groups with order divisible by 5 6 are not F SZ, and that neither are their Sylow 5-subgroups. The groups G 2 (5) and HN were previously established as non-F SZ by Peter Schauenburg; we present alternative proofs. All other sporadic simple groups and their Sylow subgroups are shown to be F SZ. We conclude by considering all perfect groups available through GAP with order at most 10 6 , and show they are non-F SZ if and only if their Sylow 5-subgroups are non-F SZ.

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