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Apr
7
comment Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups.
If you do post over on MO, be sure to cross-link the questions.
Apr
7
comment Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups.
I'd also be very interested if there is a known way to characterise a conjugation quandle. I don't recall ever having seen one.
Apr
7
comment Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups.
I don't think so (if I understand your point correctly). You could have two elements of a group that commute, but neither is a power of the other.
Apr
7
comment Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups.
I don't know if there is a complete characterisation of conjugation quandles, but conjugation quandles are crossed sets. A crossed set is a quandle that satisfies the additional condition that, for any two elements $a$ and $b$, one has $a*b = a$ if, and only if, $b*a = b$. There is a quandle of order $3$ that is not a crossed set. (Of course, it is not faithful.) However, a crossed set need not be a conjugation quandle. There are $4$ crossed sets of order $4$ (up to isomorphism), but only one conjugation quandle.
Apr
7
comment Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups.
The conjugation quandle of an abelian group is a so-called trivial quandle (i.e., x*y = x, identically), so any two abelian groups of the same order give rise to isomorphic conjugation quandles.
Mar
29
comment elementary row operations
I just meant that since the group $\operatorname{GL}(n,q)$ is finite, it contains only a finite number of elementary matrices. (An elementary row operation is just a multiplication by an elementary matrix.) Since the elementary matrices generate $\operatorname{GL}(3,2)$, all your matrices are products of them and their inverses. In general, a subgroup need not actually contain any elementary matrices (except the identity), but its elements will be products of them.
Mar
29
answered elementary row operations
Mar
27
answered A detail in Baer Theorem
Mar
25
comment Verifying $G*H \cong G' * H' \implies |G| = |G'|$ or $|G| = |H'|$ (All Groups Cyclic)
Can't you just look at the abelianisations of the groups, which must be isomorphic, and then use the fundamental theorem of finitely generated abelian groups?
Mar
24
comment Good software package to help Maths learning
Many universities make good mathematics software available to their students. Check out what your school has to offer before you spend money on something.
Mar
20
answered Maple: If a number is an integer/square
Mar
20
comment Does every finite nilpotent group occur as a Frattini subgroup?
@JackSchmidt Thank you so much for a very helpful answer! I'm just starting to read those references I can access, but it seems there is some lovely mathematics behind the answer to this question.
Mar
20
accepted Does every finite nilpotent group occur as a Frattini subgroup?
Mar
19
asked Does every finite nilpotent group occur as a Frattini subgroup?
Mar
7
comment How to prove that $N$ is 2-transitive on $\Omega$?
The reason $N$ is transitive is due to its being a normal subgroup of $G$. (Likewise for $N_{\omega}$ in $G_{\omega}$.) Being non-regular just tells you that some point stabiliser is non-trivial. (Or, that it is not transitive, but you know now that it is.)
Mar
7
comment Is the stabilizer of an element $\delta$ in the stabilizer of $\omega$ in G equal to the pointwise stabilizer of $\{ \delta, \omega \}$
Yes, this is correct.
Mar
7
answered How to prove that $N$ is 2-transitive on $\Omega$?
Feb
28
comment What should I call a partial subgroup lattice diagram?
I don't think it sounds that clunky. And, clunky or not, I'd know what you meant.
Feb
25
comment Topics in combinatorial Group Theory
The free group of rank $2$ acts on the sphere by orthogonal transformations. By carefully dissecting the free group, you can produce paradoxical decompositions. The Wikipedia article explains it all very nicely. Of course, as was already suggested, your instructor can indicate whether the topic is actually suitable for your course.
Feb
25
answered Topics in combinatorial Group Theory