Why this topic?

 embarrassed by

brute-force alternative

you hope there's a prize

Why isomorphism?

Why groups*?

*Technically a groupoid, actually a 2-, 3-, k-groupoid.

Why group isomorphism?

Main Result

Theorem A (Dietrich-W. `20+\(\varepsilon\))

There is an algorithm to decide isomorphism of solvable groups of most orders in nearly-linear time.

Before best known time \(n^{O(\log n)}\), \(n\) group order.

Nearly-linear = \(O(n^2 (\log n)^c)\).

Approximate Actual Theorem A (Dietrich-W. `20+\(\varepsilon\))

There is an algorithm that decides solvable group isomorphism on most orders in time

\[\exp O((\log\log n)^4 ).\]

Problem 1: what is "most orders"?

 

Problem 2: for \(n\gg 0\), \(\exp O((\log\log n)^4)<n\).

So algorithm is not even reading the whole group.

Actual Theorem A (Dietrich-W. `20+\(\varepsilon\))

There is a dense subset \(\Upsilon\subset\mathbb{N}\) and an algorithm such that

• Given: black-box groups \(G\) and \(\tilde{G}\) of known factored order \(n=p_1^{e_1}\cdots p_t^{e_t}\in \Upsilon\)
• Decides: if \(G\) is solvable and if so if \(G\cong \tilde{G}\).

Assuming the Extended Riemann Hypothesis (ERH), the algorithm is deterministic and runs in time \[\exp O( (\log\log n)^4 ).\]

Without ERH it can be made Las Vegas.

Actual Corollary A (Dietrich-W. `20+\(\varepsilon\))

There is a dense subset \(\Upsilon\subset\mathbb{N}\) and an algorithm such that

• Given: a pair of Cayley tables \(G\) and \(\tilde{G}\)
• Decides: if \(G,\tilde{G}\) are solvable groups with equal orders in \(\Upsilon\), and if decides if \(G\cong \tilde{G}\).

All this in time \(O(n^2( \log n)^c)\).

Related Results

Theorem B (Dietrich-W. `18 arxiv:1810.03467)

There is a dense subset \(\Psi\subset\mathbb{N}\) and an algorithm such that

• Given: black-box groups \(G\) and \(\tilde{G}\) of known factored order \(n=p_1^{e_1}\cdots p_t^{e_t}\in \Psi\)
• Decides: if \(G\) is abelian and if so if \(G\cong \tilde{G}\).

The algorithm is deterministic and runs in time \[O( (\log n)^c ).\]

 

\(\Psi\) is all \(n\) where given a prime \(p|n\), \(p^2|n\) implies \(p\leq \log\log n\) and \(p^e|n\) implies \(p^e\leq \log n\).

Call these pseudo-square-free.

Theorem B (Dietrich-W. `18 arxiv:1810.03467)

There is a dense subset \(\Psi\subset\mathbb{N}\) and an algorithm such that

• Given: black-box groups \(G\) and \(\tilde{G}\) of known factored order \(n=p_1^{e_1}\cdots p_t^{e_t}\in \Psi\)
• Decides: if \(G\) is coprime meta-cyclic ("C-groups") and if so if \(G\cong \tilde{G}\).

The algorithm is deterministic and runs in time \[O( (\log n)^c ).\]

 

See also work of Dietrich-Low arXiv:2005.02569

Theorem C (Dietrich-W. `20+\(\varepsilon\))

There is a dense subset \(\Upsilon\subset\mathbb{N}\) and an algorithm such that

• Given: black-box groups \(G\) and \(\tilde{G}\) of known factored order \(n=p_1^{e_1}\cdots p_t^{e_t}\in \Upsilon\)
• Decides: if \(G\) is nilpotent and if so if \(G\cong \tilde{G}\).

The algorithm is deterministic and runs in time \[\exp O( (\log\log n)^3 ).\]

 

\(\Psi\) is also the pseudo-square-free orders.

The Plan

Separate Concerns

Find condition on \(n\) so that

\[|G|=n\Rightarrow G=H\ltimes B\]

\(H\) = Hard group theory,

\(B\) = Bad number theory.

Intuition: if a problem can be solved it cannot be where hard group theory and bad number theory overlap.

Swap Concerns and Wiggle-your-pinky

With \[|G|=n\Rightarrow G=H\ltimes B\]

Number Theory \(\Rightarrow\) hard group theory in \(H\) is tiny.

Group Theory \(\Rightarrow\) bad number theory makes \(B\) cyclic.

Isolated Primes

(The Group Theory)

Defn.

A prime divisor \(p\) of \(n\) is isolated if given a prime-power \(q^k|n\), if \(p|(q^k-1)\) then \(k=0\).

Lemma (Dietrich-W. `18) 

For every solvable group \(G\) of order \(n\) and every isolated prime \(p|n\), \(G\) has a unique Sylow \(p\)-subgroup.

Proof.  (Hall) \(\exists P_i,~G=P_1\cdots P_t\), each \(P_i\) a Sylow \(p_i\)-sub. & \(P_iP_j=P_jP_i\).  Let \(p_u=p\).

 

(Sylow) For \(i\neq u\), #Sylow \(p\)-sub. of \(H\) is \(q^k\) with \(q^k\equiv 1 (p)\).

By condition isolation, \(k=0\).  I.e. \(P_u\lhd P_iP_u\).

 

 \(\forall g_i\in P_i\qquad g_1\cdots g_t P_u=g_1\cdots g_i P_u g_{i+1}\cdots g_t=P_u g_1\cdots g_t.\)

\(P_u\lhd G\).                                                                                      \(\Box\)

 

Defn.  A prime divisor \(p\) of \(n\) is isolated if given a prime-power \(q^k|n\), if \(p|(q^k-1)\) then \(k=0\).

Corollary (Dietrich-W. `18 arxiv:1810.03467) 

For every solvable group \(G\) of order \(n\) and

\[\pi_n=\{p: p|n, p~isolated\},\]

\(G\) has a unique nilpotent Hall \(\pi_n\)-subgroup \(B\) and a all \(\pi'_n\)-subgroup \(H\)

\[G=H\ltimes B\]

All such \(H\) are conjugate in \(G\).

Isolated primes

(The Number Theory)

Which of these is the 1000th Prime Number?

A. 7919

B. 15671

C. 50021

D. 105397

Which of these is the 1000th Prime Number?

A. 7919...think of that, nearly 1/8 numbers less than 8,000 is prime!

 

 

As primes are so abundant,

most integers factor into lots of distinct and big primes.

Theorem (Holder 1890)

A group of square-free order has a normal cyclic subgroup whose quotient is cyclic.

 

I.e. |G| square-free implies \(G\cong \mathbb{Z}/a\ltimes \mathbb{Z}/b\)

Theorem (Hardy?) 

The density of positive integers \(n\) that are square-free is at least 60%.

History. (Erdős-Pálfy `86)  

Looked a 2-way isolation condition using Brun Sieve methods, and the group theory that follows.   

 

We adapted the conditions and proofs.

Thm (Dietrich-W. `18 arxiv:1810.03467).

The set \(\Upsilon\) of pseudo-square-free integers \(n\) where each prime divisor \(p|n\) is isolated, is dense in \(\mathbb{N}\).

The Algorithm

1-AbelRecog
2-AbelRecog
FactorInt
CanonicalBasis
ExtDiscLog
IsomorphismAbelMO

IsolatedHallFindMO
IsolatedHallSplitMO
IsomphismNilpotentMO

IsolHallSplitMetaCycMO
Deconjugate

IsolHallSplitMO IsomorphismSolvableMO

Abelian

Nilpotent

Meta-cyclic

Solvable

Problems

(MO="Most Orders", i.e. \(\Psi,\Upsilon\))

1-AbelRecog
2-AbelRecog
FactorInt
CanonicalBasis
ExtDiscLog
IsomAbelMO

Abelian

30% do the obvious abelian things

70% cite deep work...

on discrete logs (Gordan, Karagiorgos-Poulakis, Teske,...), Hermite Normal Form (Havas, Lenstra-Lenstra-Lovasz, Sims,...), Number Field Sieve (Buhler-Lenstra-Pomerance,...)

Main ideas

1-AbelRecog

Given: group G

Return: isomorphism \[\alpha:\prod_i\mathbb{Z}/d_i\to G\] or prove \(\not\exists\).

IsomAbelMO

​Given: \(G,\tilde{G}\)

Decide: if \(G\) abelian and if so if \(G\cong \tilde{G}\).

ExtDiscLog

Given: basis \(x_1,\ldots,x_s\in G\) and \(g\in G\)

Return: \(g=x_1^{e_1}\cdots x_s^{e_s}\).

Use only on torsion \(\leq \log \log n\)

Trosion \(\geq \log \log n\) is cyclic

\[\alpha,\tilde{\alpha}:\prod_i\mathbb{Z}/d_i\to G,\tilde{G}\]

Like abelian, peel off small torsion leaving number theory to explain the rest is cyclic.

Main ideas

IsomNilMO

​Given: \(G,\tilde{G}\), \(|G|\in \Psi\)

Decide: if \(G\) nilpotent and if so if \(G\cong \tilde{G}\).

IsolHallFindMO

Given: \(G\), \(|G|\in \Psi\)

Return: Hall isolated prime subgroup \(B\)

Present \(G/B\) as \(B\) is recognizable.

Sylow-by-Sylow brute-force on \(H\), IsomAbelMO on \(B\).

IsolatedHallFindMO
IsolatedHallSplitMO
IsomphismNilpotentMO

Nilpotent

IsolHallSplitMO

Given: nilpotent \(G\),\(|G|\in \Psi\)

Return: \(G=H\times B\).

Take coprime powers.

\(\mathbb{Z}/a\ltimes_{\theta} \mathbb{Z}/b\) isomorphism determined by image of \(\theta\), which reduces to case \(b=q^f\) in which inside a cyclic group \(\mathbb{Z}/b^{\times}\).

Main ideas

Deconjugagte

​Given: \(G=\langle x\rangle\ltimes \langle y\rangle\), \(gcd(|x|,|y|)=1\), \(|G|\in \Upsilon\);

Return: \(k\in \mathbb{N}\) \(y^x=y^k\)

IsolHallSplitMCMO

Given: \(G\), \(|G|\in \Psi\)

Return: \(G=H\ltimes B\)

Present \(G/B\), build cyclic gen., coprime power.

Prime-by-prime reconstruct \(p\)-adic expansion of exponent \(k\).

IsolHallSplitMetaCycMO
Deconjugate

Meta-cyclic

Iteratively apply meta-cyclic method,

tricky because of need to reorder polycyclic sequence as you go (effective Jordan-Hölder).

Main ideas

IsomSolvableMO

​Given: \(G,\tilde{G}\), \(|G|\in \Psi\)

Decide: if \(G\) solvable and if so if \(G\cong \tilde{G}\).

Brute-force on \(H\), IsomAbelMO on \(B\), and deconjugate iteratively.

IsolHallSplitSolvMO
IsomSolvableMO

Solvable

IsolHallSplitMO

Given: solvable \(G\),\(|G|\in \Upsilon\)

Return: \(G=H\ltimes B\).

Multi-case applications of recursion and Meta-cyclcic

Isolated Sylow Subgroups

(General story)

Defn.

An isolated prime divisor \(p\) of \(n\) is strongly isolated if for every nonabelian simple group \(T\) with order dividing \(n\), \(p\) does not divide \(|T|\).

Theorem (Dietrich-W. `20+) 

For every group \(G\) of order \(n=2^e m\), \(m\) odd,

and every strongly isolated prime \(p|n\) with \(p>e\),

\(G\) has a unique Sylow \(p\)-subgroup.

Corollary (Dietrich-W. `20+) 

For every group \(G\) of order \(n=2^e m\), \(m\) odd, and \[\pi=\{p: p>e, p|n, p~strongly~isolated\}\]

then \(G\) has a unique nilpotent Hall \(\pi\)-subgroup \(B\) and a all \(\pi'\)-subgroup \(H\)

\[G=H\ltimes B\]

All such \(H\) are conjugate in \(G\).

Aspirational Theorem A' (Dietrich-W. `21-\(\varepsilon\))

There is an algorithm to decide isomorphism of solvable groups of most orders in nearly-linear time.

Before best known time \(n^{O(\log n)}\), \(n\) group order.

Nearly-linear = \(O(n^2 (\log n)^c)\).