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bio website halgebra.math.msu.su/staff/…
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visits member for 2 years, 4 months
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Oct
9
comment Two questions about axiomatic rank of groups
@M.Shahryari, no way. Consider the variety of abelian groups. It satisfies the metabelian law $[[x,y],[z,t]]=1$ or the nilpotency law $[[x,y],z]=1$. Do you believe that these laws are equivalent to two-variable laws?
Oct
8
comment Two questions about axiomatic rank of groups
To question 1 the answer is obviously negative. Just consider the trivial variety, it has finite axiomatic rank but satisfies all identities.
Aug
27
comment Are infinite groups “locally topologizable”?
It is on page 339 of English edition. Unfortunately, Google preview rejects to show this page to me (but I may read the table of contents).
Aug
26
comment Are infinite groups “locally topologizable”?
I have a Russian edition. This is Chapter 10, Section 31, Subsection 3.
Aug
25
comment Invariant planes of a nilpotent matrix with two Jordan blocks of size two
This is the description of all of them, because any invariant subspace containing a vector $v$ shoud contain $Nv$.
Aug
25
comment Invariant planes of a nilpotent matrix with two Jordan blocks of size two
Take any vector $v$ not belonging to the kernel of $N$ and consider the space $\langle v,Nv\rangle$. This family and the kernel of $N$ are all spaces you ask about.
Aug
23
comment A sequence of subsets of an infinite group
A related question: mathoverflow.net/q/179177/24165 .
Apr
24
comment An element $g$ in a group such that neither $g=1$ nor $g\ne 1$ can be proved.
@Dan, surely. What you said is basically Metatheorem 3.5.2 from my answer.
Jan
20
comment Why the axiomatic rank of the variety of groups is equal to three?
Your octonion argument is much simpler and works perfectly. The algebra of all unit octonions or all non-zero octonions (with operations $\cdot$, ${\ }^{-1}$, and $1$) is non-associative but satisfies all two-variable laws that are consequences of the group axioms.
Jan
17
comment Almost uniquely generated groups
Oh yes, thank you!
Jan
17
comment Almost uniquely generated groups
"Any finite $p$-group which is relatively free in some variety has this property by the same argument. In light of Geoff's answer perhaps these are the only ones?" -- No. The quaternion group of order 8 is not relatively free but satisfies the conditions.
Jan
17
comment Almost uniquely generated groups
Why $\{a_{1},b_{1},g_{2},g_{3}, \ldots, g_{n} \}$ is irredundant?
Jan
17
comment Almost uniquely generated groups
No. $S$ is unique up to automorphisms; so, the complement of $S$ is not necessary the Frattini subgroup.Take a cyclic group of prime order, for example.
Jan
16
comment Almost uniquely generated groups
Yes. And probably you mean Question 2 as you use that the orders are finite.
Jan
16
comment Almost uniquely generated groups
This condition is indeed weaker. See the (wrong) answer of M. Shahryari.
Jan
16
comment Almost uniquely generated groups
No. Even $F_1$ does not satisfy. The question is about inclusion-minimal sets.
Jan
16
comment Almost uniquely generated groups
It seems that you use the letter $n$ in two different senses. You are talking about $m$-generated free groups in the varieties of nilpotent groups of exponent $p^n$, right?
Jan
14
comment Minimal generating sets of groups
Actually, any (inclusion-)minimal generating set of a Tarski monster consists of two elements (if $p$ is prime).
Jan
9
comment relatively free groups in $Var(S_3)$
Yes, exactly. By the way ,the group of polynomial-with-coefficients functions is also free but in a variety of different universal algebras --- a variety of groups with marked elements (i.e., groups with additional 0-ary operations.
Jan
9
comment relatively free groups in $Var(S_3)$
@M.Shahryari, see the edit. I guess that 324 is the number of one -variable polynomial functions over $S_3$ in a different sense of the word polynomial; propably they mean polynomials with coefficients from the group --- such as $x(12)x^2(123)$