## Finite Group and Subgroup Criteria

## Problem 160

Let $G$ be a finite group and let $H$ be a subset of $G$ such that for any $a,b \in H$, $ab\in H$.

Then show that $H$ is a subgroup of $G$.

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of the day

Let $G$ be a finite group and let $H$ be a subset of $G$ such that for any $a,b \in H$, $ab\in H$.

Then show that $H$ is a subgroup of $G$.

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Let $G$ be a non-abelian simple group. Let $D(G)=[G,G]$ be the commutator subgroup of $G$. Show that $G=D(G)$.

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Let $K, N$ be normal subgroups of a group $G$. Suppose that the quotient groups $G/K$ and $G/N$ are both abelian groups. Then show that the group

\[G/(K \cap N)\]
is also an abelian group.

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Let $G$ be a group and let $D(G)=[G,G]$ be the commutator subgroup of $G$.

Let $N$ be a subgroup of $G$.

Prove that the subgroup $N$ is normal in $G$ and $G/N$ is an abelian group if and only if $N \supset D(G)$.

Let $G$ be a finite group of order $n$ and let $m$ be an integer that is relatively prime to $n=|G|$. Show that for any $a\in G$, there exists a unique element $b\in G$ such that

\[b^m=a.\]

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Let $G$ and $H$ be groups and let $f:G \to K$ be a group homomorphism. Prove that the homomorphism $f$ is injective if and only if the kernel is trivial, that is, $\ker(f)=\{e\}$, where $e$ is the identity element of $G$.

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Let $\R^{\times}=\R\setminus \{0\}$ be the multiplicative group of real numbers.

Let $\C^{\times}=\C\setminus \{0\}$ be the multiplicative group of complex numbers.

Then show that $\R^{\times}$ and $\C^{\times}$ are not isomorphic as groups.

Let $G$ be a group and $H$ and $K$ be subgroups of $G$.

For $h \in H$, and $k \in K$, we define the commutator $[h, k]:=hkh^{-1}k^{-1}$.

Let $[H,K]$ be a subgroup of $G$ generated by all such commutators.

Show that if $H$ and $K$ are normal subgroups of $G$, then the subgroup $[H, K]$ is normal in $G$.

Add to solve laterLet $G$ be a nilpotent group and let $H$ be a subgroup such that $H$ is a subgroup in the center $Z(G)$ of $G$.

Suppose that the quotient $G/H$ is nilpotent.

Then show that $G$ is also nilpotent.

Add to solve later Let $G$ be a group. Suppose that $H_1, H_2, N_1, N_2$ are all normal subgroup of $G$, $H_1 \lhd N_2$, and $H_2 \lhd N_2$.

Suppose also that $N_1/H_1$ is isomorphic to $N_2/H_2$. Then prove or disprove that $N_1$ is isomorphic to $N_2$.

Let $p$ be a prime number.

Let $G$ be a non-abelian $p$-group.

Show that the index of the center of $G$ is divisible by $p^2$.

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Let $G$ be an infinite cyclic group. Then show that $G$ does not have a composition series.

Add to solve laterLet $G$ be a finite group. Then show that $G$ has a composition series.

Add to solve laterLet $G$ be a finite group of order $18$.

Show that the group $G$ is solvable.

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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 $N_G(H)=H$.

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Let $G$ and $G’$ be groups and let $f:G \to G’$ be a group homomorphism.

If $H’$ is a normal subgroup of the group $G’$, then show that $H=f^{-1}(H’)$ is a normal subgroup of the group $G$.

Let $A$, $B$ be groups. Let $\phi:B \to \Aut(A)$ be a group homomorphism.

The semidirect product $A \rtimes_{\phi} B$ with respect to $\phi$ is a group whose underlying set is $A \times B$ with group operation

\[(a_1, b_1)\cdot (a_2, b_2)=(a_1\phi(b_1)(a_2), b_1b_2),\]
where $a_i \in A, b_i \in B$ for $i=1, 2$.

Let $f: A \to A’$ and $g:B \to B’$ be group isomorphisms. Define $\phi’: B’\to \Aut(A’)$ by sending $b’ \in B’$ to $f\circ \phi(g^{-1}(b’))\circ f^{-1}$.

\[\require{AMScd}

\begin{CD}

B @>{\phi}>> \Aut(A)\\

@A{g^{-1}}AA @VV{\sigma_f}V \\

B’ @>{\phi’}>> \Aut(A’)

\end{CD}\]
Here $\sigma_f:\Aut(A) \to \Aut(A’)$ is defined by $ \alpha \in \Aut(A) \mapsto f\alpha f^{-1}\in \Aut(A’)$.

Then show that

\[A \rtimes_{\phi} B \cong A’ \rtimes_{\phi’} B’.\]

Let $G$ be a simple group and let $X$ be a finite set.

Suppose $G$ acts nontrivially on $X$. That is, there exist $g\in G$ and $x \in X$ such that $g\cdot x \neq x$.

Then show that $G$ is a finite group and the order of $G$ divides $|X|!$.

Let $X$ be a subset of a group $G$. Let $C_G(X)$ be the centralizer subgroup of $X$ in $G$.

For any $g \in G$, show that $gC_G(X)g^{-1}=C_G(gXg^{-1})$.

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Let $\F_p$ be the finite field of $p$ elements, where $p$ is a prime number.

Let $G_n=\GL_n(\F_p)$ be the group of $n\times n$ invertible matrices with entries in the field $\F_p$. As usual in linear algebra, we may regard the elements of $G_n$ as linear transformations on $\F_p^n$, the $n$-dimensional vector space over $\F_p$. Therefore, $G_n$ acts on $\F_p^n$.

Let $e_n \in \F_p^n$ be the vector $(1,0, \dots,0)$.

(The so-called first standard basis vector in $\F_p^n$.)

Find the size of the $G_n$-orbit of $e_n$, and show that $\Stab_{G_n}(e_n)$ has order $|G_{n-1}|\cdot p^{n-1}$.

Conclude by induction that

\[|G_n|=p^{n^2}\prod_{i=1}^{n} \left(1-\frac{1}{p^i} \right).\]