Articles of group theory

Need help about $P\Gamma L_2(q)$, $q=4,3$

I am asking kindly, For which values of $n$ we have $$S_n≅P\Gamma L_2(3),S_n≅P\Gamma L_2(4)$$ This may be correct if we replace $S_n$ by $A_n$. Any help will be appreciated. 🙂 Edit (JL): Adding the definition of the group. Let $\Omega$ denote the set $GF(q)\cup\{\infty\}$. Then the group $P\Gamma L_2(q)$ consists of bijections from $\Omega$ to […]

The product of all the elements of a finite abelian group

I’m trying to prove the following statements. Let $G$ be a finite abelian group $G = \{a_{1}, a_{2}, …, a_{n}\}$. If there is no element $x \neq e$ in $G$ such that $x = x^{-1}$, then $a_{1}a_{2} \cdot \cdot \cdot a_{n} = e$. Since the only element in $G$ that is an inverse of itself […]

Are there cube-free numbers $n$, for which the number of groups of order $n$ is unknown?

For squarefree $n$, there is a formula allowing to compute the number of groups of order $n$. I do not think that such a formula exists for cubefree numbers. If a cubefree number $n$ has the property, that there is no prime power $p^k|n$, such that $p^k\equiv 1\ (\ mod\ q\ )$ for some prime […]

Prove that $gNg^{-1} \subseteq N$ iff $gNg^{-1} = N$

I’m reading the solution to one of my homework problems and am stuck on something. Here is the problem: Let $N \leq G$ be a finite subgroup. Show that $gNg^{-1} \subseteq N$ if and only if $gNg^{-1} = N$. The solution is to suppose that $gNg^{-1} \subseteq N$ and consider the map $\varphi: N \to […]

Adapting a proof on elements of order 2: from finite groups to infinite groups

Consider the following problem, appropriate for a first course in Group Theory: Problem: Prove that there cannot be a group with exactly two elements of order $2$. General Proof: Suppose for the sake of contradiction that there are exactly two elements of order $2$, and denote them by $a$ and $b$. Note $ab \neq a, […]

Infinitely many simple groups with conditions on order?

If $G$ is a non abelian finite simple group, we know that $4$ divides $|G|$. More precisely there are infinitely many finite simple groups $G$ such that $v_2(|G|)=2$, just consider $\mathrm{PSL}_2(\mathbb F_p)$ with $p\equiv 3 \pmod 8$, $p$ prime and $p>3$. So my question tries to generalize the above statement: Do we know the primes […]

Any two groups of three elements are isomorphic – Fraleigh p. 47 4.25(b)

The answer has no details. Hence maybe the answer is supposed to be quick. But I can’t see it? Hence I took two groups. Call them $G_1 = \{a, b, c\}, G_2 = \{d, e, f\}$. Then because every group has an identity, I know $G_1, G_2$ has one each. Hence WLOG pick $c$ as […]

Showing that a homomorphism between groups of units is surjective.

This question already has an answer here: Why does the natural ring homomorphism induce a surjective group homomorphism of units? 2 answers

Character on conjugacy classes

Let $V_j$, $j = 1,2$ be finite dimensional representations of a group $G$. Show: $\chi_{V_j}$ is a constant on each conjugacy class of $G$, where $\chi_{V_j}$ is the character of the representation. I’ve just started with group theory and have a really hard time so I’d like someone to confirm what I did so far […]

Commutator Identities in Groups

Let $x$, $y$, $z$ be elements of a group $G$ and let $[x,y]=x^{-1}y^{-1}xy$ be the commutator of $x$ and $y$. Then we have the following identities: $[x,zy]=[x,y][x,z][[x,z],y]$ $[xz,y]=[x,y][[x,y],z][z,y]$ My question is, is there any identity for $[x_1x_2\cdots x_m, y_1y_2\cdots y_n]$, generating the identities above? It may not be difficult but I did not find it […]