# 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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### On direct sum and direct product of groups

I understood the definitions of direct sums and direct products of groups (rings, modules, etc). The definitions of them can be given in terms of universal properties - certain commutative diagram in ...
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### Determine abelian groups with 48 elements

I am just doing some revision for my linear algebra exam, and I came across this problem: Determine all abelian groups (up to isomorphism) with exactly 48 elements. I am not sure I have ever ...
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### For prime $p$, let $G$ be a group such that every non-identity element of $G$ has order $p$. Show that if $|G|$ is finite, then $|G| = p^n$.

I've been self teaching myself some topics in preparation for university and thought I'd have a go at some past paper questions from their website. As such I do not have much experience with these ...
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### Why are Lie groups automatically analytic manifolds?

In the book by Kolar, Michor, and Slovak, it is shown that multiplication $\mu:G\times G\to G$ is analytic in some neighborhood of $e$. Specifically, they show that in the chart given by $\exp^{-1}$, ...
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### What is $pA$ when $p$ is a prime number and $A$ an abelian group?

Let $A$ be a finite abelian $p$-group. I want to prove that $pA$ is also an abelian finite $p$-group, of order strictly less than the order of $A$. The problem is that I don't even know what does the ...
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### Proving Sylow theorems without using group actions

Most of the proofs of Sylow theorems involves groups actions in some way as below: Sylow theorems - wiki There is a thread for them here, too: Proofs of Sylow theorems. However, I would like to see ...
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### real irreducible representation of SU(2) group

Consider a real irreducible representation of SU(2) group in d-dimensional space-time. How many components do the spinors (eigenvectors) have ? For instance, a real irreducible spinor in 10-dim has 16 ...
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### Subgroups of $(\mathbb{Z}\times \mathbb{Z}) * \mathbb{Z}/2\mathbb{Z}$

I can't figure out who the group $(\mathbb{Z}\times \mathbb{Z}) * \mathbb{Z}/2\mathbb{Z}$ is. (Where $*$ means the free product). In particular I would like to study all its subgroups. Is it true ...
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### Problem with the proof of abelian finite groups decomposition

I can't understand the proof which says that every finite abelian $p$-group can be written as a direct sum of cyclic $p$-groups. I'm using Lang's book of Algebra. My problem is about the following ...
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### Surjective group homomorphism $f:A_4\times \mathbb{Z}_2 \to S_3$ [closed]

Is there a surjective group homomorphism $f:A_4\times \mathbb{Z}_2 \to S_3$? I need to find one if the answer is "yes" or explain why the answer is "no". With $\mathbb{Z}_n$ I mean the quotient ...
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### Explicitly calculating the group of global isometries of a convex regular polygon (dihedral group)

Instead of an intuitive geometric description of the dihedral groups $D_{2n}$, that one can find in virtually every good book on group theory, I want to calculate the global isometries of a convex ...
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### Cyclic subgroup proof question

Let $a$ be an element of order $n$ in a group and let $k$ be a positive integer. Then $\langle a^k \rangle = \langle a^{\gcd(n,k)} \rangle \text{ and } |a^k| = n / \gcd(n,k).$ The proof starts by ...
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### Prove /Disprove: Existence of Symmetric Subgroup of Order $n^2-1$ of $S_n$.

$G$ is a symmetric subgroup of symmetric group $S_n$ acting on $n$ objects where order of $G$ is $n^2-1$. Problem: Prove (or disporve), Such $G$ exists for $S_n$ for finite $n$. For $n=5,11,71$, ...
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### Commensurability for vector spaces

Let me start by saying I am a not a mathematician and I have not studied group theory (just a few brushes here and there) but after reading I have a very basic understanding of commensurability as ...
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### Determining a group given some elements

Say we have a group G and we know some of the elements (but not all). How does one determine the order and list all the elements of the group in an intuitive way? In this case G is the smallest ...
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### Order of factor group

Question: Determine the order of $(\mathbb{Z} \times \mathbb{Z})/ \left<(4,2)\right>$. Is the group cyclic? I want to first apologize for the way this post is written. I'm on the road and ...
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### Subgroup of Order $n^2-1$ in Symmetric Group $S_n$ when $n=5, 11, 71$

$G$ is a symmetric subgroup of symmetric group $S_n$ acting on $n$ objects where $n=5, 11, 71$ and order of $G$ is $n^2-1$. Question: Does $G$ (as defined above for $n=5, 11, 71$) exist? How can I ...
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### For a prime $p$ if $p^m = p^n+2\cdot p^k$ then $p=3$.

I read an article on commuting graphs of groups and at some point, author gets the equality $|\langle x,Z\rangle| = |Z|\cup 2\cdot |x^G|$ where $Z$ denotes the center of the $p$-group $G$ and $x^G$ ...
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### Finite number of orbit type for compact Lie group actions

For the linear SO(3) actions, the number of orbit types (or the number of isotropy class) is finite : this seem to be a classical result coming from Bredon (Introduction to compact transformation ...
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### If $G/N\cong H$ then $G=NH$?

I am curious if $G/N\cong H$ then $G=NH$? ($N$ is a normal subgroup of $G$, and $H$ is a subgroup of $G$.) With this setup, we get that $NH$ is a subgroup of $G$ so $NH\subset G$. I am not sure ...
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### Two equivalent conditions proof (related to semiproduct of groups)

Let $G$ be a group, and let $N$ be a normal subgroup of $G$, $H$ any subgroup of $G$. I wish to prove the equivalence of (i) $G$ is the product of subgroups, $G=NH$, where $N\cap H=\{e\}$. (ii) ...
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### solve a specific word problem in free groups

Let $F_2=\langle a, b\rangle$ be the non-abelian free group with two generators and $e$ is the neutral element in $F_2$. Given $g\in F_2, k\geq 2$ an integer. I want to know how to solve the word ...
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### Sylow tower theorem involving supersolvable groups

I just want to find out if anyone has a reference to the result that states that if $G$ is a finite supersolvable group then it has a normal Sylow subgroup.
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### Prove that L(x) = (x - 1)/p is a discrete logarithm function in a group.

I have the following problem: Let $p$ be a prime number and $G$ be a set of all $x\in \mathbb{Z}_{p^2}$, such that $x \equiv 1 \pmod{p}$. Prove that: $G$ is a multiplicative group (regarding ...
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### soft question- Kenneth's Brown notation at “Cohomology of finite Groups”

As the title indicates, my question has to do with something rather simple. So, in Kenneth's Brown book "Cohomology of finite groups" at pg.84-85 and in particular Theorem 10.3 and Proposition 10.4, ...
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### Reference for results p-adic integers Z_p as abelian group

I have two facts I want to use in my thesis about $\mathbb{Z}_p$. To be precise: automorphism group is $\mathbb{Z}_p \times \mathbb{Z}/(p-1)\mathbb{Z}$, except for 2, and that any subgroup with finite ...
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### Finding a 3-embedded subgroup.

I have the group of order 108 $G=(((\mathbb{Z}_3 \times \mathbb{Z}_3)\ltimes \mathbb{Z}_3)\ltimes \mathbb{Z}_2) \ltimes \mathbb{Z}_2$ obtained from an algorithm in GAP, but I need to prove that it has ...
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### Product with a normal subgroup

If $H \unlhd K \unlhd G$ and $P\unlhd G$ then does necessarily $HP \unlhd KP$? I can see this is true using the correspondence theorem since $HP/P \unlhd KP/P$ I want to try direct and prove it ...
If two groups $G\bigoplus \mathbb{Z}$ and $H\bigoplus \mathbb{Z}$ are commensurable. Does it imply that the groups $G$ and $H$ are commensuarble?
Let $G$ be a finite group such that all subgroups of prime-power order are pronormal in $G$. If $M$ is a normal $p$-subgroup of $G$ then all prime-power order subgroups of $G/M$ are pronormal in $G/M$....