# Tagged Questions

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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### Does $\forall v ( T_1 v = 0 \lor T_2 v = 0 \lor \dots \lor T_n v =0 )$ imply $T_1 = 0 \lor T_2 = 0 \lor \dots \lor T_n = 0$?

Let $V$ and $W$ be vector spaces and $T_1$, $T_2$, $\dots$, $T_n$ be linear transformations from $V$ to $W$, such that for every $v$ in $V$, either $T_1 v = 0$, $T_2 v = 0$, $\dots$ or $T_n v = 0$. ...
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### Groups of order $pq$ have a proper normal subgroup

I am doing the following exercise from [Birkhoff and MacLane, A survey of modern algebra]: Let $G$ be a group of order $pq$ ($p,q$ primes). Show that either $G$ is cyclic or contains an element ...
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### A simple group such that $[G:H]=n$ can be embedded into $A_n$

Let $G$ be a finite simple group and $H$ be a proper subgroup of $G$ such that $|G:H|=n$. Then, how do I prove that $G$ can be embeded into $A_n$? I can prove that $G$ can be embedded into $S_n$ ...
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### Have semigroups with actions on themselves that have a dual to the compatibility axiom ever been studied?

For a semigroup $G$ with a left action on itself, the axiom for compatibility becomes: $$\forall f,g,h\in G:hg(f)=h(g(f))$$ Now suppose there is additional axiom, or constraint if you prefer, ...
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### Maximum number of Sylow subgroups

I've been studying Sylow-$p$ subgroups, and I've come across this problem. Let $G$ be a finite group. Show that the number of Sylow subgroups of $G$ is at most $\frac{2}{3}|G|$ . ($|G|$ is the ...
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### How to show every subgroup of a cyclic group is cyclic?

I'm teaching a group theory course now, and I wanted to give my students a proof that every subgroup of a cyclic group is cyclic. The easiest way I could think to do this is to say that any cyclic ...
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### Recent advancement in Haar measure

From my personal interest I have studied Haar Measure and the related concept of group theory on my own. However due to the lack of an authoritative source it is not getting possible for me to know ...
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### Conjugacy classes in $D_4$

Let G be group of all symmetries of square. Find number of conjugate classes in G. I tried this question just as we do for $S_n$ that the number of conjugate classes in $S_n$ is partition number of ...
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### On the maximum number of Sylow subgroups

I have some doubts regarding the first sentences below the fourth case in the more elementary solution by Prof. Samuel to problem 11856 of the Monthly. Here you have a screenshot of the relevant part ...
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### Why if $(\mathbf B, \cdot)$ is a finite order group with prime order then $(\mathbf B, \cdot)$ is cyclic? [duplicate]

In the notes I'm studying from ( again =) ) I read: If $(\mathbf B, \cdot)$ is a finite order group with prime order then $(\mathbf B, \cdot)$ is cyclic Could someone give me a justification for ...
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### Need help in understanding a certain step of a certain proof in finite group theory and group actions

A proof is from Aluffi's textbook "Algebra: Chapter 0". A statement: There are no simple groups of order $24$. The proof from the book: Let $G$ be a group or order $24 = 2^33$, and consider ...
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### Orbits of the permutation action of a subgroup on its cosets

Consider a finite group $G$ and a subgroup $H \subseteq G$. There is a transitive group action of $G$ on the set of left cosets $gH$ by left multiplication, and the stabilizer of $gH$ is $gHg^{-1}$. ...
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### Prove that $G$ is cyclic if $|G|=15$ and $G$ has only one subgroup each of orders $3$ and $5$

Question: Let $\left | G \right |=15$. If G has only one subgroup of order 3 and only one subgroup of order 5, prove that G is cyclic. Looking for useful hints to the above question. Thanks in ...
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### $G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then $G=HK$?

Let $G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then is it true that $G=HK$ ? ( I know that the fact is true if $p=2$ ...
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### Order of product of non-disjoint cycles

Let $a$ and $b$ be two non-disjoint cycles of order $m$ and $n$. Is there any general formula for the order of $a b$? I understand that we can convert any non-disjoint cycles into disjoint cycles and ...
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### Some questions about a proof referring to hyperbolic group

I feel confused about: 1:It says that "otherwise H contains an infinite cyclic characteristic subgroup C," however, by definition, if H is elementary, we can only get that H contains an infinite ...
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Let $A$ be a finitely generated abelian group and $\alpha:A\to A$ be an endomorphism. Since $A=A_{free}\oplus A_{torsion}$, we can induce $\bar \alpha:A_{free}\to A_{free}$, i.e. $\bar\alpha$ is a map ...
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### Proving that $A+B - (A \cap B) = A \cup B$ for binary integers

I hope computing questions are fine here. I'm trying to show that for all binary numbers $A$ and $B$, $A+B - (A \cap B) = A \cup B$. It's confusing me firstly because I'm not sure what the "set ...
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Let G be a nonempty set closed under an associative product,which in addition satisfies: (1) There exists an $e\in G$ such that $a.e=a \forall a \in G$ (2)Give $a \in G$, there exists an element $y(... 2answers 60 views ### Aut$(K/F)$permutes roots of polynomial. Let Aut$(K/F)$is the set of all automorphism from$F$to$K$, where$K$is a galois extension of$F$. Let$f(x) \in F[x]$and$\alpha$be a root of the polynomial$f(x)$. I am able to prove that for ... 1answer 621 views ### General linear group/special linear group is isomorphic to R* Let$GL(n,\mathbb{R})$be the group of invertible$n \times n$real matrices, let$SL(n,\mathbb{R})$be the group of$n \times n$real matrices of determinant$1$, and$\mathbb{R}^*$be the group of ... 2answers 43 views ### If$gh = hg, \ \ \gcd(|g|, |h|) = 1$, then$|gh| = |g||h|$($|a|$is the order of element$a$in a group$G$) Let$G$be a group and$g,h \in G$. I need to prove that if$g$and$h$commute and their orders are coprime, then$|gh| = |g||h|$, that is, the order of their product is the multiple of their orders. ... 3answers 48 views ### Ley$G$be a group of prime order$p$. Then$|Aut(G)|=p-1$Let$G$be a group of order$p$where$p$is a prime number( hence,$G$is cyclic ) Prove that the group of automorphisms of$G$has order$p-1$. Since$p$is prime, for any homomorphism$\phi: G \to ...
If I have an abelian group $G$ of order $p^n$, how can I decide if it's isomorphic to $\Bbb{Z}_p \times \Bbb{Z}_p \times\ldots \times \Bbb{Z}_p$ ($n$ times) or to \$\Bbb{Z}_{p^2} \times \Bbb{Z}_p \...