## 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 $4$.

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Sylow’s Theorem Problems and Solutions.

Check out the post “Sylow’s Theorem (summary)” for the statement of Sylow’s theorem and various exercise problems about Sylow’s theorem.

The other popular posts in Group Theory are:

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 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 finite group of order $217$.

**(a)** Prove that $G$ is a cyclic group.

**(c)** Determine the number of generators of the group $G$.

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.

Suppose that $G$ is a finite group of order $p^an$, where $p$ is a prime number and $p$ does not divide $n$.

Let $N$ be a normal subgroup of $G$ such that the index $|G: N|$ is relatively prime to $p$.

Then show that $N$ contains all $p$-Sylow subgroups of $G$.

Add to solve laterLet $G$ 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.

Read solution

Let $G$ be a finite group and $P$ be a nontrivial Sylow subgroup of $G$.

Let $H$ be a subgroup of $G$ containing the normalizer $N_G(P)$ of $P$ in $G$.

Then show that $N_G(H)=H$.

Read solution

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

(a) $|G|=100$.

(b) $|G|=200$.

Read solution

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 $r$.

Read solution

**(a)** Show that if a group $G$ has the following order, then it is not simple.

- $28$
- $496$
- $8128$

**(b) **Show that if the order of a group $G$ is equal to an even * perfect number* then the group is not simple.

In this post we review Sylow’s theorem and as an example we solve the following problem.

Show that a group of order $200$ has a normal Sylow $5$-subgroup.

Read solution

Let $p$ be a prime number. Suppose that the order of each element of a finite group $G$ is a power of $p$. Then prove that $G$ is a $p$-group. Namely, the order of $G$ is a power of $p$.

Add to solve later