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16h
comment $2$-groups with odd permutations
Another hint. Try the case in which $H=1$ first; then generalise.
1d
comment Verifying proof that set of all group homomorphisms is an abelian group
Yes, you need to show closure. If you are writing your group $H$ additively, why do you get $f(u)-f(u)=1$ rather than $0$?
Oct
23
answered Surprising identities / equations
Oct
23
answered properties on groups of order $p^2qr$
Oct
23
answered Number of different magmas up to isomorphism
Oct
22
comment Normalizer and centralizer are equivalent when $p$ is the smallest prime dividing $|G|$
@Akaichan, I've edited with the additional information.
Oct
22
revised Normalizer and centralizer are equivalent when $p$ is the smallest prime dividing $|G|$
Added more information.
Oct
22
revised Normalizer and centralizer are equivalent when $p$ is the smallest prime dividing $|G|$
Added more information.
Oct
22
answered Normalizer and centralizer are equivalent when $p$ is the smallest prime dividing $|G|$
Oct
21
awarded  Nice Question
Oct
20
comment How to build a subgroup $H\leq S_4$ having order $8$?
The Klein $4$-group has order $4$! You might try starting with a $4$-cycle such as $(1,2,3,4)$ and attempt to build up a dihedral group.
Oct
20
answered Covering finite groups by unions of proper subgroups
Oct
16
comment Do solvable groups have elementary abelian characteristic subgroups?
I see now; you are specifically taking the $N_i$ to be $p$-groups for a single $p$.
Oct
16
comment Do solvable groups have elementary abelian characteristic subgroups?
What if $N_1$ and $N_2$ are elementary abelian for different primes, such as in the dihedral group of order $20$? Perhaps not the best example, since those individual minimal normals are themselves characteristic. Could it occur that there are $N_1$ and $N_2$ for different primes where neither $N_i$ is characteristic?
Oct
16
comment Do solvable groups have elementary abelian characteristic subgroups?
Technically, the trivial subgroup works. But I think you can grab a non-trivial one out of the last non-trivial term of the derived series.
Oct
16
answered Do solvable groups have elementary abelian characteristic subgroups?
Oct
14
answered To show a finite group G is nilpotent
Oct
13
revised Calculating the Order of An Element in A Group
Improved formatting a bit.
Oct
13
answered Calculating the Order of An Element in A Group
Oct
13
comment Show that the order of $G$ cannot be divisible by $7$.
Maybe it means "not symmetric or alternating"? Those seem to be the only two primitive (=transitive) groups of degree $19$ with order divisible by $7$.