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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 I need example to satisfy this lemma: Let $P$ be a $p$-group and let $N$ be a nontrivial
If you take $X = \left\langle\left(\array{1 1\\0 1}\right)\right\rangle$ and $N=\mathbb{F}_p^2$ for $p>2$ then you get a counterexample for your lemma by taking $y=(1,1)$.
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".
Jan
21
answered Direct product of simple groups
Jan
20
comment Direct product of simple groups
As the projection is onto, the image of a normal subgroup is normal.
Jan
20
comment Direct product of simple groups
@DerekHolt: He just tries to prove that $L$ is isomorphic to $H_1$ or $H_2$, not that it is $H_1$ or $H_2$, so the statement is OK.
Dec
27
revised Group of order 30 can't be simple
The "corrected" solution was faulty, so I removed the wrong part.
Dec
27
suggested approved edit on Group of order 30 can't be simple
Dec
25
comment Group of order 30 can't be simple
How do you get in (2) that 5-Sylow subgroups are normal if $s=6$?
Dec
23
comment Group of order 30 can't be simple
Part (3) of your postscript surprises me a bit. Since when is 6 a prime number?
Dec
22
comment Group of order 30 can't be simple
@Nishant: I wanted to point out to you that Thompson's transfer lemma is another possible generalization of this line of argument, but then the first hit on google was this. You get the conclusion of your comment by taking $k=1$ in your question from July (looks a bit odd, but that's probably the reason you didn't make your observation back then).
Dec
20
comment Group of order $p^2$ is commutative with prime $p$
Try to use first Lemma 2 and then Lemma 1.
Dec
20
comment Group of order $p^2$ is commutative with prime $p$
Sylow's theorems are more commonly used for groups that are not $p$-groups.