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Kourovka
Kourovka
2.81(a,b)
Finite first-order recognition of
F
(
X
)
F(X)
F
(
X
)
and
Z
(
X
)
\mathbb Z^{(X)}
Z
(
X
)
from subsemigroup lattices
Kourovka
3.12
A residually finite, locally finite group containing
S
5
S_5
S
5
with a full
{
2
,
3
,
5
}
\{2,3,5\}
{
2
,
3
,
5
}
-Sylow basis
Kourovka
3.20
First-order separation of pro-orderable groups from a bi-orderable matrix group over
Z
[
F
2
]
\mathbb Z[F_2]
Z
[
F
2
]
Kourovka
3.43
Cardinal-length cyclic ascent and radical separation
R
κ
(
G
κ
)
=
1
<
R
μ
(
G
κ
)
R_\kappa(G_\kappa)=1<R_\mu(G_\kappa)
R
κ
(
G
κ
)
=
1
<
R
μ
(
G
κ
)
Kourovka
4.17
Finite traces of group varieties via
F
n
/
⋂
ker
f
F_n/\bigcap\ker f
F
n
/
⋂
ker
f
and residual quotient closure
Kourovka
4.40
Canonical images of
B
⋊
A
B\rtimes A
B
⋊
A
in
A
×
Aut
(
B
C
⋊
C
)
A\times\operatorname{Aut}(B^C\rtimes C)
A
×
Aut
(
B
C
⋊
C
)
and ambient conjugacy
Kourovka
4.55
Nonunique rank-
88
88
88
projective decompositions over
Z
(
11
)
[
PSL
2
(
F
11
)
]
\mathbb Z_{(11)}[\operatorname{PSL}_2(\mathbb F_{11})]
Z
(
11
)
[
PSL
2
(
F
11
)]
Kourovka
5.38
Eventual centrality of surjective endomorphisms and Hopficity of
A
×
B
A\times B
A
×
B
for finitely generated soluble groups
Kourovka
5.39
Faithful actions
C
↪
Aut
(
S
)
C\hookrightarrow\operatorname{Aut}(S)
C
↪
Aut
(
S
)
with
d
(
S
)
≤
5
d(S)\le5
d
(
S
)
≤
5
and
S
(
4
)
=
1
S^{(4)}=1
S
(
4
)
=
1
Kourovka
6.9
Countable internal central factors
G
′
=
⟨
H
i
⟩
G^\prime=\langle H_i\rangle
G
′
=
⟨
H
i
⟩
in arbitrary locally normal groups
Kourovka
7.15
An involution of
2
⋅
A
8
2\!\cdot\!A_8
2
⋅
A
8
with central fixed involutions and nonquaternion Sylow
2
2
2
-subgroups
Kourovka
7.38(a)
A nonnilpotent pronilpotent variety
Pro
(
C
)
\operatorname{Pro}(\mathcal C)
Pro
(
C
)
with only nilpotent locally nilpotent subvarieties
Kourovka
7.58
Coordinate retractions and the image of
F
/
V
(
R
)
↪
B
⋊
(
F
/
R
)
F/\mathcal V(R)\hookrightarrow B\rtimes(F/R)
F
/
V
(
R
)
↪
B
⋊
(
F
/
R
)
Kourovka
8.16
Central chief factors of
Z
2
+
⋊
Z
2
×
\mathbb Z_2^+\rtimes\mathbb Z_2^\times
Z
2
+
⋊
Z
2
×
and failure of normal closure for
Z
‾
\overline Z
Z
-groups
Kourovka
8.54(b)
All-arity identities of two-generated metabelian groups via
(
n
,
I
)
(n,I)
(
n
,
I
)
in
Z
[
X
±
1
,
Y
±
1
]
\mathbb Z[X^{\pm1},Y^{\pm1}]
Z
[
X
±
1
,
Y
±
1
]
Kourovka
8.74
Central graphs and failure of intersection closure for good subnormal subgroups of
E
×
V
E\times V
E
×
V
Kourovka
9.4
Finite separating isotopes of
G
m
!
+
1
G^{m!+1}
G
m
!
+
1
and non-generation of isotope closures
Kourovka
9.41(c)
A proper subgroup
P
<
Sym
(
N
)
P<\operatorname{Sym}(\mathbb N)
P
<
Sym
(
N
)
matching all countable partitions into infinite parts
Kourovka
9.47
Full Vietoris representations in
N
(
G
)
\mathcal N(G)
N
(
G
)
, clopen principal ideals, and an
ω
1
\omega_1
ω
1
-triangle obstruction
Kourovka
10.4
Metacyclic
p
p
p
-groups of order
p
4
k
+
4
p^{4k+4}
p
4
k
+
4
with unbounded class and nonordinary mixed varieties
Kourovka
10.43
A nonstandard rational permutation representation
P
E
3
(
Z
)
→
GL
7
(
Q
)
PE_3(\mathbb Z)\to\operatorname{GL}_7(\mathbb Q)
P
E
3
(
Z
)
→
GL
7
(
Q
)
Kourovka
10.44
The standard inclusion
E
3
(
F
3
)
↪
E
4
(
F
3
)
E_3(\mathbb F_3)\hookrightarrow E_4(\mathbb F_3)
E
3
(
F
3
)
↪
E
4
(
F
3
)
without a natural-module semilinear isomorphism
Kourovka
10.52
Normalized root words and reflective completion
G
^
=
l
i
m
←
G
/
P
n
(
G
)
\widehat G=\varprojlim G/P_n(G)
G
=
lim
G
/
P
n
(
G
)
with exact power filtration
Kourovka
11.5
A nonfree projective
(
F
2
×
F
2
)
[
1
]
(\mathbb F_2\times\mathbb F_2)[1]
(
F
2
×
F
2
)
[
1
]
-module containing two independent elements
Kourovka
11.15
Total computable two-variable laws
a
p
,
n
a_{p,n}
a
p
,
n
detecting abelian Sylow
p
p
p
-subgroups for
n
≥
∣
G
∣
n\ge|G|
n
≥
∣
G
∣
Kourovka
14.22
An irreducible centralizer over
Z
2
⋊
M
Z
\mathbb Z^2\rtimes_M\mathbb Z
Z
2
⋊
M
Z
whose radical has no finite normal isolated basis
Kourovka
15.89
A four-regular bipartite Cayley graph with point spectrum
C
∖
{
0
}
\mathbb C\setminus\{0\}
C
∖
{
0
}
on
C
G
\mathbb C^G
C
G
Kourovka
16.9
Palindromic length in
F
n
F_n
F
n
via reflection length in
C
2
∗
(
n
+
1
)
C_2^{*(n+1)}
C
2
∗
(
n
+
1
)
and noncrossing matchings
Kourovka
16.28(a)
A closed
Y
⊆
GL
3
(
F
5
(
u
)
‾
)
Y\subseteq\operatorname{GL}_3(\overline{\mathbb F_5(u)})
Y
⊆
GL
3
(
F
5
(
u
)
)
with
P
∈
Y
m
‾
∖
Y
m
P\in\overline{Y^m}\setminus Y^m
P
∈
Y
m
∖
Y
m
for every
m
≥
2
m\ge2
m
≥
2
Kourovka
17.34
Divisible subgroups as exact equalizers and trivial dominions in torsion-free nilpotent groups of class
≤
c
\le c
≤
c
Kourovka
17.101
Faithful inner-homogeneous
R
R
R
-representations via HNN extensions with cardinal bound
max
{
ℵ
0
,
∣
R
∣
,
∣
G
∣
,
∣
V
∣
}
\max\{\aleph_0,\allowbreak|R|,\allowbreak|G|,\allowbreak|V|\}
max
{
ℵ
0
,
∣
R
∣
,
∣
G
∣
,
∣
V
∣
}
Kourovka
17.124
Primitive-recursive certificates for
G
P
′
′
=
1
G_P^{\prime\prime}=1
G
P
′′
=
1
and effective enumeration via Laurent relations
Kourovka
18.11
Embeddings
∗
i
∈
I
G
i
↪
Aut
(
F
∑
i
n
i
+
∣
I
∣
+
1
)
\ast_{i\in I}G_i\hookrightarrow\operatorname{Aut}(F_{\sum_i n_i+|I|+1})
∗
i
∈
I
G
i
↪
Aut
(
F
∑
i
n
i
+
∣
I
∣
+
1
)
from separating cocycles
Kourovka
18.76
A nonsplit extension
1
→
D
+
→
E
→
F
2
×
F
2
→
1
1\to D^+\to E\to F_2\times F_2\to1
1
→
D
+
→
E
→
F
2
×
F
2
→
1
with the left-multiplication action
Kourovka
21.40
A torsion-free normal
U
≤
G
≤
GL
n
(
Q
)
U\le G\le\operatorname{GL}_n(\mathbb Q)
U
≤
G
≤
GL
n
(
Q
)
with
γ
n
(
U
)
=
1
\gamma_n(U)=1
γ
n
(
U
)
=
1
and
[
G
:
U
]
≤
(
2
n
+
1
)
d
[G:U]\le(2n+1)^d
[
G
:
U
]
≤
(
2
n
+
1
)
d
Kourovka
21.68
A semiabelian group of order
2592
2592
2592
with an irreducible nonmonomial representation of degree
8
8
8
Kourovka
21.106
A positive parameter-free formula defining
{
c
1
,
c
−
1
}
\{c_1,c_{-1}\}
{
c
1
,
c
−
1
}
in the residually finite integer Heisenberg group
Kourovka
21.121(a)
Nonattainment of the
p
p
p
-Jordan exponent:
E
p
(
Γ
)
=
(
1
,
∞
)
\mathcal E_p(\Gamma)=(1,\infty)
E
p
(
Γ
)
=
(
1
,
∞
)
for a triangular locally finite group
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