Be $G$ a group of order $35,$ Prove at least $1$ Element lies not in a subgroup of order $7$
158799
Order of $G$ is 35, are there elements of order 5 and 7 in G?
26398
Group of even order contains an element of order 2
104112
Number of finite simple groups of given order is at most $2$ - is a classification-free proof possible?
58673
Elements of order 5 in $S_7$, odd permutations of order 4 in $S_4$, and find a specific permutation in $S_7$
100613
Prove that if an average of a thousand numbers is less than 7, then at least one of the numbers being averaged is less than 7
15526
Question on groups of order $pq$
90329
Prove there is no simple group of order $729$
89508
Existence of normal subgroups for a group of order $36$
157853
$Q_8$ is isomorphic to a subgroup of $S_8$, but not isomorphic to a subgroup of $S_n$ for $n\leq 7$.
15633
If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.
44152
If $G$ has only one subgroup of order $n$, then that Subgroup is Normal
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
157841
Existence of a non-abelian group of order $p^n$.
47239
For $p,q$ primes such that $q|p-1$, show there is only one non-abelian group of order $pq$, up to isomorphism
123711
Cyclic group with exactly 3 subgroups: itself $\{e\}$ and one of order $7$. Isn't this impossible?
15210
Group of order 6 proof
134676
Number of elements of order $p$ is a multiple of $p-1$ (finite group).