## 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 laterLet $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$.

Add to solve laterLet $G$ 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 $Z(G)$.

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 laterProve that any $p$-Sylow subgroup of a group $G$ of order $33$ is a normal subgroup of $G$.

Add to solve laterLet $p, q$ 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.

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 $G$.

Let $G$ be a finite group of order $18$.

Show that the group $G$ is solvable.

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Determine whether a group $G$ of the following order is simple or not.

(a) $|G|=100$.

(b) $|G|=200$.

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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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