11,461 reputation
11451
bio website coolnumbers.wordpress.com
location Everywhere
age
visits member for 2 years, 6 months
seen 3 hours ago

.


Feb
18
revised Which groups have the property that $HK$ is a subgroup for all noncyclic subgroups $H$ and $K$?
added 21 characters in body
Feb
18
revised Which groups have the property that $HK$ is a subgroup for all noncyclic subgroups $H$ and $K$?
added 927 characters in body
Feb
18
answered Which groups have the property that $HK$ is a subgroup for all noncyclic subgroups $H$ and $K$?
Feb
16
answered $xT=x$ iff $x=e$ implies that every $g$ may be written as $x^{-1}(xT)$ for some $x\in G$.
Feb
15
comment finite odd order abelian group property
Also, the $x$ is unique.
Feb
14
answered Show that $S_4/V$ is isomorphic to $S_3 $, where $V$ is the Klein Four Group.
Feb
11
answered On the center of a finite group $G$ with a normal Sylow subgroup
Feb
11
comment On the center of a finite group $G$ with a normal Sylow subgroup
But none of these examples have a normal Sylow $p$-subgroup.
Feb
11
revised Example of group homomorphism $f: G \to G$ that is injective, but not surjective.
edited tags
Feb
11
answered Example of group homomorphism $f: G \to G$ that is injective, but not surjective.
Feb
11
comment Is there a simple and a non-simple group with same numbers of elements of each order.
I don't know if there is a book that deals with this stuff, but I doubt it. But there are many articles about the subject, and questions on this site and MO. There are some properties you can deduce from the number of elements of each order. Suppose $G$ and $H$ have the same number of elements of each order. You can show that if $G$ is nilpotent, then $H$ must be nilpotent as well. Also, if $G$ is supersolvable, then $H$ must be solvable. I think it is still an open question whether $G$ solvable implies $H$ solvable when $G$ and $H$ have the same number of elements of each order.
Feb
11
comment What About The Converse of Lagrange's Theorem?
See also this question. In terms of $d$, we cannot do any better than Sylow's theorem. If $d$ is not a power of a prime, then there exists a group of order divisible by $d$ but with no subgroup of order $d$. In terms of $G$, the groups that satisfy the converse of Lagrange's theorem are called CLT groups. You can show that supersolvable groups are CLT and that CLT groups are solvable.
Feb
11
awarded  Custodian
Feb
11
reviewed Reject suggested edit on The prime numbers that divide $10^4-1$
Feb
11
answered Is there a simple and a non-simple group with same numbers of elements of each order.
Feb
10
revised If $p$ is an odd prime, does every Sylow $p$-subgroup contain an element not in any other Sylow $p$-subgroup?
added 143 characters in body
Feb
10
asked If $p$ is an odd prime, does every Sylow $p$-subgroup contain an element not in any other Sylow $p$-subgroup?
Feb
9
answered $G$ is a finite group and $G_1$,$G_2$ are subgroup that $G_1 \cap G_2=\{1\}$, then $|G_1|.|G_2| \big| |G|$?
Feb
8
awarded  Enlightened
Feb
8
awarded  Nice Answer