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Jan
21
revised Fantastic properties of Z/2Z
added 12 characters in body
Jan
21
answered Fantastic properties of Z/2Z
Jan
21
answered Examples of cancellative normal semigroups
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
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 $Var(S_3)$