Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4
Problem 566
Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.
Add to solve later Tagged: normal Sylow subgroup
Add to solve later Every Group of Order 72 is Not a Simple Group
Every Sylow 11-Subgroup of a Group of Order 231 is Contained in the Center $Z(G)$
Problem 464
Let $G$ be a finite group of order $231=3\cdot 7 \cdot 11$.
Prove that every Sylow $11$-subgroup of $G$ is contained in the center $Z(G)$.
Add to solve later Non-Abelian Group of Order $pq$ and its Sylow Subgroups
Problem 293
Let $G$ be a non-abelian group of order $pq$, where $p, q$ are prime numbers satisfying $q \equiv 1 \pmod p$.
Prove that a $q$-Sylow subgroup of $G$ is normal and the number of $p$-Sylow subgroups are $q$.
Add to solve later Sylow Subgroups of a Group of Order 33 is Normal Subgroups
Problem 278
Prove that any $p$-Sylow subgroup of a group $G$ of order $33$ is a normal subgroup of $G$.
Add to solve later Group of Order $pq$ Has a Normal Sylow Subgroup and Solvable
Problem 245
Let $p, q$ be prime numbers such that $p>q$.
If a group $G$ has order $pq$, then show the followings.
(a) The group $G$ has a normal Sylow $p$-subgroup.
(b) The group $G$ is solvable.
Add to solve later If a Sylow Subgroup is Normal in a Normal Subgroup, it is a Normal Subgroup
Problem 226
Let $G$ be a finite group. Suppose that $p$ is a prime number that divides the order of $G$.
Let $N$ be a normal subgroup of $G$ and let $P$ be a $p$-Sylow subgroup of $G$.
Show that if $P$ is normal in $N$, then $P$ is a normal subgroup of $G$.
Add to solve later Group of Order 18 is Solvable
Problem 118
Let $G$ be a finite group of order $18$.
Show that the group $G$ is solvable.
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Add to solve later Are Groups of Order 100, 200 Simple?
Problem 100
Determine whether a group $G$ of the following order is simple or not.
(a) $|G|=100$.
(b) $|G|=200$.
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Add to solve later A Group of Order $pqr$ Contains a Normal Subgroup of Order Either $p, q$, or $r$
Problem 81
Let $G$ be a group of order $|G|=pqr$, where $p,q,r$ are prime numbers such that $p<q<r$.
Show that $G$ has a normal subgroup of order either $p,q$ or $r$.
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