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Jan
19 |
answered | Non finitely based varieties of groups defined by finitely many variables |
Jan
17 |
comment |
Almost uniquely generated groups
Oh yes, thank you! |
Jan
17 |
awarded | Citizen Patrol |
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
16 |
asked | Almost uniquely generated groups |
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 $x(12)x^2(123)$ |
Jan
9 |
revised |
relatively free groups in $Var(S_3)$
added 1235 characters in body |
Jan
8 |
answered | relatively free groups in $Var(S_3)$ |
Dec
17 |
comment |
Large abelian characteristic subgroups in abelian-by-countable groups
Dear Yves, if you do not mind, we shall include your example in our preprint (op. cit.) with a proper reference of course. As for the Podoski--Szegedy example, it is maybe simpler but we do not know how to convert it to the normal-but-not-characteristic settings. Their example is (in essence) the subgroup $\hbox{Alt}(\mathbb Z)A$ of the uncountable symmetric group $\hbox{Sym}(\mathbb Z)$, where $\hbox{Alt}(\mathbb Z)$ is the alternating group (consisting of even finitary permutations) and $A$ is the elementary abelian 2-group consisting of permutations preserving absolute values of integers. |
Dec
17 |
accepted | Large abelian characteristic subgroups in abelian-by-countable groups |
Dec
15 |
comment |
Large abelian characteristic subgroups in abelian-by-countable groups
Oh, yes. I mixed up finitely supported matrices and matrices $f$ with finite $\hbox{rank}(f-1)$ (i.e operators $f$ such that $\hbox{ker}(f-1)$ is of finite codimension). The latter form a normal subgroup but the former do not. Inside $H$, these two subgroups coincides, right? (I mean, their intersections with $H$ coincides.) |
Dec
15 |
comment |
Large abelian characteristic subgroups in abelian-by-countable groups
Excuse me, Yves. I do not understand your comment. The finitely supported matrices form a normal subgroup of $G$, right? If they do, then the commutator must be finitely supported. |