# 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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### Show that $U(8)$ is Isomorphic to $U(12)$.

Question: Show that $U(8)$ is Isomorphic to $U(12)$ The groups are: $U\left ( 8 \right )=\left \{ 1,3,5,7 \right \}$ $U\left ( 12 \right )=\left \{ 1,5,7,11 \right \}$ I think there is a bit of ...
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### How to find the invariant forms of a finite group

Let $G\subset GL(n,\mathbb{Z})$. I am looking for an algorithm that finds all symmetric matrices $F$ left invariant by G, ie $$g^TFg=F\quad \forall g\in G.$$ I have found lists of these invariants for ...
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### Sylow subgroup of a symmetric group

Consider the symmetric group of$S_{20}$ and it's subgroup $A_{20}$ consisting of all even permutations. Let $H$ be a $7$-Sylow subgroup of$A_{20}$. Is $H$ cyclic? And is correct the statement which ...
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### Existence of a $G$-invariant metric on a principal bundle

Given a smooth principal $G$-bundle $\pi: P \rightarrow M$, I want to show the existence of connections by showing that $P$ admits $G$-invariant metrics. I was thinking in a kind of averaging ...
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### Not getting how to prove reverse hypothesis.

This is a theorem from Dummit & Foote text- Let $G$ be a group acting on the non-empty set $A$.The relation on $A$ defined by $a \sim b$ iff $a=g.b$ for some $g \in G$ is an ...
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### Converse of Lagrange Theorem for p-groups.

I need clarification on my understanding of Sylow Theorems. Can I say that a finite p-group will have a subgroup for each prime power? If the above is valid, can I say then that p-groups satisfy the ...
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### Simple groups and irreducible characters of degree 3

The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}_5$ and $PSL(2,7)$. That seems to be a result coming from Blichfelt's work on $GL(3,\mathbb{C})$, which I ...
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### Condition in a theorem of Hall

There is a well-celebrated theorem of Hall, which characterizes solvable groups according to the existence of Hall-$\pi$ subgroups. In this theorem, I was wondering whether it can be stated in a ...
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### If $G$ isn't abelian and $|G|=p^{3}$ then $Z(G)=G'$

Let $G$ a finite group and $G' \subseteq G$ the smallest normal subgroup of $G$ such that $G/G'$ is abelian. Prove that if $G$ isn't abelian and $|G|=p^{3}$ then $Z(G)=G'$ My attempt: If $G$ is not ...
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### If $H$ and $G/H$ are $p$-groups then $G$ is a $p$-group.

Please verify: If $H$ is a $p$-group, then $|H| = p^r$, for some integer $r$. If $G/H$ is a $p$-group, then $|G/H| = p^s$, for some integer $s$. But the cardinality of a quotient set is the index, ...
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### Groups of order $25$

Please verify my solution that there are only two groups of order $25$ up to isomorphism. As $|G|$ is a prime squared, then $G$ is abelian. Since the Theorem of Finite Abelian Groups, $G$ is a direct ...
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### Are there at least $3$ groups of order $16$ that has element of order $8$?

Are there at least $3$ groups of order $16$ that has element of order $8$? I know that probably the simplest way of doing this problem is looking at the element structure of the abelian groups of ...
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### Clarification on a concept involving Hall subgroups

Suppose that $G$ is a finite group and $H\leq G$. If $Q \in$ Hall$_{\pi}(G)$ such that $Q \cap H \in$ Hall$_{\pi}(H)$. Is true that $Q \leq H$ when Hall$_\pi(H)$ $\subseteq$ Hall$_\pi(G)$
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### If C/A is abelian, then C/B is abelian.

I am trying to prove that if groups A $\lhd$ B $\lhd$ C, A $\lhd$ C, and C/A is abelian, then C/B and B/A are abelian. Clearly, B/A is abelian since it is a subgroup of the abelian group C/A. ...
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### Proof/Reasoning why the sgn function which counts inversions has the following property?

$\mathrm{sgn}(\pi\circ\sigma)=\mathrm{sgn}(\pi)\cdot \mathrm{sgn}(\sigma)$ I am familiar with how to count inversions and any insight for why this formula holds true would be very helpful.
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### Seeing composition table

Let p > 2 be any prime number. The non-zero integers modulo p form a group of order p - 1 with respect to multiplication modulo p. $$[a][b] = [ab] = ab (mod p)$$ denoted Gp. The identity is ...
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### generators of $\mathbb{Z}_p^*$ are all the elements in $\mathbb{Z}_p^*$?

I know that a finite group with a prime number of elements is cyclic and every element in the group is a generator for the group. Thinking about $(\mathbb{Z}_p^*, \cdot)$ I thought that the order of ...
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### Group theory finding proper subgroup [closed]

The smallest order for a group to have a non-abelian proper subgroup? I am confused how shall I proceed pls help Thanks in advance
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### How to find a subgroup of the external direct product $\Bbb{Z}/4\Bbb{Z}\oplus\Bbb{Z}/2\Bbb{Z}$ not of the form $H\oplus K$?

How to find a subgroup of the external direct product $\Bbb{Z}/4\Bbb{Z}\oplus\Bbb{Z}/2\Bbb{Z}$ not of the form $H\oplus K$, where $H$ and $K$ are subgroups of $\Bbb{Z}/4\Bbb{Z}$ and $\Bbb{Z}2/\Bbb{Z}$,...
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### Suppose φ : G → G is a group homomorphism, and H is the image of G under φ. If G is of polynomially bounded growth, show that H is too.

Suppose φ : G → G is a group homomorphism, and H is the image of G under φ. If G is of polynomially bounded growth, show that H is too. [the question is above as well as in the title] how would I ...
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### A natural example of f.g. group, with uncountably many normal subgroups, but is not SQ-universal

I am looking for "natural" examples of a finitely generated group which have uncountably many normal subgroups but are not SQ-universal. A group $G$ is SQ-universal if for any countable group $H$, $H$ ...
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### Prove that FG is of exponential growth where F is any ﬁeld and G is the free group on {x_1, . . . , x_n}. [closed]

Prove that FG is of exponential growth where F is any ﬁeld and G is the free group on {x_1, . . . , x_n}. where FG is: Given a group G and a ﬁeld F, we can combine them into a group algebra. Setwise, ...
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### Why is this set not a group?

So the group $Z_8 - \{0\}$ does not form a group under multiplication modulo $8$. The reason being is because its not closed as $2 \cdot 4 = 0$. I understand what closure is. If $a$ and $b$ are in ...
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### $G \cong H$ and $G$ is simple. Then $H$ is simple as well.

I think it must be true. Yet I have no rigorous proof for that. So, what I need to prove is that "group being simple" is invariant under isomorphism. That if $G \cong H$, then either both are simple ...
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### How can we show $gnu(8892)=gnu(9324)$ by hand?

With GAP it can be verified immediately that there are $303$ groups of order $8892=2^2\cdot 3^2\cdot 13\cdot19$ and also $303$ groups of order $9324=2^2\cdot3^2\cdot 7\cdot 37$. I do not expect that ...
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Let $G$ be a finite group. It is easy to show that $G'\le \bigcap_{\chi\in Irr(G)}Kerdet\chi$. Is there equality ? This question arises from the remarkable equalities $\bigcap_{\chi\in Irr(G)}Ker\... 0answers 43 views ### Determinant of a character let two characters$\chi$and$\vartheta$of a finite group$G$(assumed to be non-null). Let$\mathfrak{X}$and$\mathfrak{Y}$be representations of$G$affording respectively$\chi$and$\vartheta$... 2answers 42 views ### Are the groups$G:=\{A \in M(n,\mathbb R) : A=A^t\}$i.e. the group ( under addition ) of symmetric matrices and$O(n,\mathbb R)$isomorphic? Let$G:=\{A \in M(n,\mathbb R) : A=A^t\}$i.e. the group ( under addition ) of symmetric matrices ; Are$G$and$O(n,\mathbb R)$isomorphic ? 2answers 26 views ### A simple group such that$[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$... 0answers 40 views ### Recent advancement in Haar measure From my personal interest I have studied Haar Measure and the related concept of group theory on my own. However due to the lack of an authoritative source it is not getting possible for me to know ... 2answers 29 views ### Why if$(\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 ... 2answers 19 views ### 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 ... 1answer 45 views ### 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 ... 2answers 31 views ###$G$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$p=2$... 1answer 53 views ### A proposition about free product My question is about a proposition that I found on Munkres book: Topology (page 419). My definitions are based on this book. I give you the definition of free product that he uses.$\textbf {...
We color a equilateral triangle by coloring each edge with one of $k \geq 1$ colors. Find a formula for the number of orbits under the action of $D_6$, the dihedral group of $6$ elements, on the ...
### 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 $T_n v = 0$. ...