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 Category: Group Theory
Add to solve later Eckmann–Hilton Argument: Group Operation is a Group Homomorphism
Problem 268
Let $G$ be a group with the identity element $e$ and suppose that we have a group homomorphism $\phi$ from the direct product $G \times G$ to $G$ satisfying
\[\phi(e, g)=g \text{ and } \phi(g, e)=g, \tag{*}\]
for any $g\in G$.
Let $\mu: G\times G \to G$ be a map defined by
\[\mu(g, h)=gh.\]
(That is, $\mu$ is the group operation on $G$.)
Then prove that $\phi=\mu$.
Also prove that the group $G$ is abelian.
Add to solve later Equivalent Definitions of Characteristic Subgroups. Center is Characteristic.
Problem 246
Let $H$ be a subgroup of a group $G$. We call $H$ characteristic in $G$ if for any automorphism $\sigma\in \Aut(G)$ of $G$, we have $\sigma(H)=H$.
(a) Prove that if $\sigma(H) \subset H$ for all $\sigma \in \Aut(G)$, then $H$ is characteristic in $G$.
(b) Prove that the center $Z(G)$ of $G$ is characteristic in $G$.
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 Pullback Group of Two Group Homomorphisms into a Group
Problem 244
Let $G_1, G_1$, and $H$ be groups. Let $f_1: G_1 \to H$ and $f_2: G_2 \to H$ be group homomorphisms.
Define the subset $M$ of $G_1 \times G_2$ to be
\[M=\{(a_1, a_2) \in G_1\times G_2 \mid f_1(a_1)=f_2(a_2)\}.\]
Prove that $M$ is a subgroup of $G_1 \times G_2$.
Add to solve later A Group Homomorphism is Injective if and only if Monic
Problem 243
Let $f:G\to G’$ be a group homomorphism. We say that $f$ is monic whenever we have $fg_1=fg_2$, where $g_1:K\to G$ and $g_2:K \to G$ are group homomorphisms for some group $K$, we have $g_1=g_2$.
Then prove that a group homomorphism $f: G \to G’$ is injective if and only if it is monic.
Add to solve later No Finite Abelian Group is Divisible
Problem 240
A nontrivial abelian group $A$ is called divisible if for each element $a\in A$ and each nonzero integer $k$, there is an element $x \in A$ such that $x^k=a$.
(Here the group operation of $A$ is written multiplicatively. In additive notation, the equation is written as $kx=a$.) That is, $A$ is divisible if each element has a $k$-th root in $A$.
(a) Prove that the additive group of rational numbers $\Q$ is divisible.
(b) Prove that no finite abelian group is divisible.
Add to solve later Subgroup of Finite Index Contains a Normal Subgroup of Finite Index
Problem 232
Let $G$ be a group and let $H$ be a subgroup of finite index. Then show that there exists a normal subgroup $N$ of $G$ such that $N$ is of finite index in $G$ and $N\subset H$.
Add to solve later Subgroup Containing All $p$-Sylow Subgroups of a Group
Problem 227
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 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 Cyclic Group if and only if There Exists a Surjective Group Homomorphism From $\Z$
Problem 225
Show that a group $G$ is cyclic if and only if there exists a surjective group homomorphism from the additive group $\Z$ of integers to the group $G$.
Add to solve later Group of $p$-Power Roots of 1 is Isomorphic to a Proper Quotient of Itself
Problem 221
Let $p$ be a prime number. Let
\[G=\{z\in \C \mid z^{p^n}=1\} \]
be the group of $p$-power roots of $1$ in $\C$.
Show that the map $\Psi:G\to G$ mapping $z$ to $z^p$ is a surjective homomorphism.
Also deduce from this that $G$ is isomorphic to a proper quotient of $G$ itself.
Add to solve later Use Lagrange’s Theorem to Prove Fermat’s Little Theorem
Problem 219
Use Lagrange’s Theorem in the multiplicative group $(\Zmod{p})^{\times}$ to prove Fermat’s Little Theorem: if $p$ is a prime number then $a^p \equiv a \pmod p$ for all $a \in \Z$.
Add to solve later If Every Nonidentity Element of a Group has Order 2, then it’s an Abelian Group
Problem 212
Let $G$ be a group. Suppose that the order of nonidentity element of $G$ is $2$.
Then show that $G$ is an abelian group.
Add to solve later Group Homomorphism, Conjugate, Center, and Abelian group
Problem 209
Let $G$ be a group. We fix an element $x$ of $G$ and define a map
\[ \Psi_x: G\to G\]
by mapping $g\in G$ to $xgx^{-1} \in G$.
Then prove the followings.
(a) The map $\Psi_x$ is a group homomorphism.
(b) The map $\Psi_x=\id$ if and only if $x\in Z(G)$, where $Z(G)$ is the center of the group $G$.
(c) The map $\Psi_y=\id$ for all $y\in G$ if and only if $G$ is an abelian group.
Add to solve later Group Homomorphism, Preimage, and Product of Groups
Problem 208
Let $G, G’$ be groups and let $f:G \to G’$ be a group homomorphism.
Put $N=\ker(f)$. Then show that we have
\[f^{-1}(f(H))=HN.\]
Add to solve later A Group Homomorphism and an Abelian Group
Problem 207
Let $G$ be a group. Define a map $f:G \to G$ by sending each element $g \in G$ to its inverse $g^{-1} \in G$.
Show that $G$ is an abelian group if and only if the map $f: G\to G$ is a group homomorphism.
Add to solve later Order of the Product of Two Elements in an Abelian Group
Problem 205
Let $G$ be an abelian group with the identity element $1$. Let $a, b$ be elements of $G$ with order $m$ and $n$, respectively.
If $m$ and $n$ are relatively prime, then show that the order of the element $ab$ is $mn$.
Add to solve later Two Normal Subgroups Intersecting Trivially Commute Each Other
Problem 196
Let $G$ be a group. Assume that $H$ and $K$ are both normal subgroups of $G$ and $H \cap K=1$. Then for any elements $h \in H$ and $k\in K$, show that $hk=kh$.
Read solution
Add to solve later Abelian Normal Subgroup, Intersection, and Product of Groups
Problem 195
Let $G$ be a group and let $A$ be an abelian subgroup of $G$ with $A \triangleleft G$.
(That is, $A$ is a normal subgroup of $G$.)
If $B$ is any subgroup of $G$, then show that
\[A \cap B \triangleleft AB.\]
Add to solve later 
