Let order of a group be $108$. Show that there exist a normal subgroup of order $27$ or $9$.
65972
Show that a group of order $108$ has a normal subgroup of order $9$ or $27$.
90449
Showing that a Sylow subgroup of a group of order $p^{2}q$ is normal.
118022
Show that a group of order $180$ is not simple.
8841
A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
46940
On Groups of Order 315 with a unique sylow 3-subgroup .
76943
Existence of subgroup of order six in $A_4$
141458
How many subgroups or order 8 an abelian Group of order 72 can have
53409
Show group of order $4n + 2$ has a subgroup of index 2.
145127
For any integers $m,n>1$ , does there exist a group $G$ with elements $a,b \in G$ such that $o(a)=m , o(b)=n$ but $ab$ has infinite order ?
115204
Show that SO(n) is a normal subgroup of O(n)
116786
normal subgroups of Alternating group
126113
Prove the existence of order 4 subgroups of order 8 groups
43058
If $n > 2$, prove that the order of the multiplicative group of units modulo n, $U_n$, is even.
44152
If $G$ has only one subgroup of order $n$, then that Subgroup is Normal
79055
Orders of a symmetric group
133001
$G$ simple group and there exists a subgroup of index $n$. Show that: $|G|$ divides $n!/2$