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Feb
23
comment What's so special about the group axioms?
You are not the only one not recognizing the importance of these axioms at once. Historically seen it took more than a century before they were generally considered important in mathematics. Shortly ago I attended a lecture by Bernd Fischer (discoverer of three sporadic finite simple groups) who spoke about his first years in university as student. Several objects (like the sympletic group) were introduced to him without even mentioning that they were groups.
Feb
23
comment A group whose automorphism group is cyclic
I'd read the original question as speaking of automorphisms as an abelian group. In your example the two automorphisms are the only ones that additionally preserve the structure as an abelian variety. There are surely more group automorphisms that do not preserve this structure.
Feb
20
comment What does this statement mean?
$\psi\circ\phi$ does not have to be onto, so if you want to identify its image (which is a set of cosets in $H$) with some set of cosets in $G$ you have to restrict the image, not the domain of the map. The "is" in the mysterious last sentence shouldn't be taken literally, but instead as "can be identified with".
Feb
20
comment How many non-isomorphic central extensions of a cyclic group of order $2$ by the Lamplighter group exist?
Very nice, thank you. I'm still struggling to understand if the extra-special $2$-groups $N_n$ are isomorphic or not (if you happen to know, a simple yes/no would be appreciated a lot), but the $G_n$ are different.
Feb
19
comment How many non-isomorphic central extensions of a cyclic group of order $2$ by the Lamplighter group exist?
@pjs36: Sorry for driving you nuts. Earlier I didn't have time to look up "\wr" and just copied and pasted from the other question.
Feb
19
comment Finitely many group extensions?
@DerekHolt: You're right. As you still seem to think that your construction works, I'll post it as a question.
Feb
19
comment Can a normal subgroup of a finite nonabelian group be nonabelian?
$D_{4n}$ has $D_{2n}$ as subgroup of index $2$ and subgroups of order $2$ are normal. Even easier is to take the direct product of two non-abelian groups.
Feb
19
comment Finitely many group extensions?
@DerekHolt: As the extensions proposed in your comment have to be central, how do you hope to put a central element of order $2$ "below" the Lamplighter Group? (I have a hard time finding a second possible $G$.)
Feb
18
comment Finitely many group extensions?
@DerekHolt: You can distinguish the groups by checking if all elements of order $2^n$ are central or not.
Feb
18
comment Finitely many group extensions?
If you take $F=Z_3$ and pick different homomorphisms from $H$ to $Aut(F)$ (only the $k$th summand of $H$ being not in the kernel) then the proof that the groups are not isomorphic follows easily from looking at the central elements of $G$.
Feb
17
comment Converse of Lagrange's Theorem
Do you know the subgroups of $A_4$?
Jan
29
comment Why is Multiplicative Notation Used for Groups (Instead of Additive)?
Before groups were known to be interesting on their own, permutation groups had been known for quite a while. As these are (invertible) functions on a set, the composition is written as $f\circ g$ or even $fg$ leading to a multiplicative notation.
Jan
26
comment Group of order 396 isn't simple
Alternatively one can look at the centralizer of an element of order $3$ in the normalizer of an $11$-Sylow subgroup to get a subgroup of order 99 rsp. index 4.
Jan
23
comment Direct product of simple groups
@daPollak: I forgot to mention that you might need to prove that the image is normal in the first line of my answer (in case your lecture didn't cover it) and the "it is easy to show" might need a very short proof, too.
Jan
22
comment Online Archive of Master Thesis
The inverse Galois problem is surely an interesting topic. If the thesis is supposed to do more than just compile a list of all known results and give some introduction to the used techniques, things might become quite tough. Take for example a look at "Groups as Galois Groups: An Introduction" by Helmut Volklein.
Jan
21
comment Characterizing the Prüfer $p$-group
Is $G$ also in 2. supposed to be infinite abelian (maybe with all proper subgroups finite)?
Jan
21
comment Non-isomoprhic semidirect products and their centers
@Leppala: You are on the right track. Also in the cyclic case the solution depends on whether $p = 1$ mod $4$ or not.
Jan
21
comment Direct product of simple groups
Exchanging $H_1$ and $H_2$ one sees that the case $H_1 \cap L = 1 = H_2 \cap L$ can only occur when $H_1$ and $H_2$ are isomorphic. Myself's answer shows that both have to be additionally abelian.
Jan
21
comment Hardness of discrete log in additive group
For finite fields with small characteristic the discrete logarithm happens to be less difficult than thought before, see A quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic.
Jan
21
comment Hardness of discrete log in additive group
Somehow you forgot to mention that the "discrete log" in the additive group is otherwise known as "division".