# Tagged Questions

Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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### 2 question on solvable group's property

A theorem says a finite group G is solvable if and only if for every divisor n of $\vert G\vert$ such that (n,$\frac{\vert G\vert}{n})=1$,G has a subgroup of order n. Does this imply that G must ...
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### how is the jordan-holder theorem used in conjunction with short exact sequences to construct groups of certain order?

I am an undergraduate and have been asked to explain how simple groups can be used to construct groups of finite order. I started with reading about the extension problem in group theory and from ...
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### How to prove that every nonabelian group of odd order is not simple?

How to prove or disprove that every nonabelian group of odd order is not simple? I have no ideas concerning that.
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### Groups of order $n^2$ with no subgroup of order $n$ [duplicate]

Is it possible to classify those groups whose order is $n^2$ for some natural number $n$ but which do not have any subgroups of order $n$? To be a bit more specific (in case a full classification is ...
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### Groups of order $n^2$ that have no subgroup of order $n$

For which $n$ is there a group of order $n^2$ without a subgroup of order $n$. Such groups can not be nilpotent. This question is related to Sudokus as composition tables of finite groups.
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### N is a normal subgroup of G if $aNa^{-1} \subset N$ for all $a ∈ G$. Prove that in that case, $aNa^{-1} = N$.

I said if N is a normal subgroup of G when $aNa^{-1} \subset N$ aN = Na as N is a normal subgroup of $G$. Therefore $aNa^{-1} = Naa^{-1}$ and $aNa^{-1} = N$. I would like to go with this proof ...
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### Structure of a group, $G$, of order $pq$ where $p, q$ are prime.

There is a proposition in Beachy and Blair's Abstract Algebra that I don't entirely follow. The proposition is the following: Let $G$ be a group of order $pq$, where $p > q$ are primes. a) If ...
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### Counting apartments in spherical buildings

Is there a formula for the number of apartments in a finite, spherical building? To be specific, is there a formula that depends on the associated Coxeter group and the thickness of the building? ...
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### An elementary question regarding a multiplicative character over finite fields

Reading Chapter 2 of Koblitz's Introduction to Elliptic Curves and Modular Forms, I got stuck on the following question. I would like to proceed my reading, so I would appreciate any hint to this. I ...
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### If $G$ is a non-cyclic group of order $n^2$, then $G$ is isomorphic to $\mathbb{Z_n} \oplus \mathbb{Z_n}$

I've independently come up with a question (I know it's been asked before, but I can't find the question online) involving the external direct product, non-cyclic groups and isomorphisms. So, is the ...
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### Relation of order of a permutation with its sign

Let $G$ be a group with order $2m$ where $m$ is odd. Consider the left action $\lambda_g:G\to G$. It appears that if $g$ has odd order iff $\lambda_g$ has odd order iff $\lambda_g$ is an even ...
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### pemutation representation that confuses me a lot recently

For any group G, define group action on a set A. There will be a permutation representation of that group action. I am kind of confused why the permutation representation can be used to reflect the ...
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### Show that Icosahedral group does not have normal subgroup of order 5

Here is my work so far: Let $G$ denote the group of orientation-preserving isometries of Icosahedron. I have shown that $G$ acts transitively on $G/G_s$ where $G_s$ is the isotropy group, namely ...
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### Normal subgroup created by a bunch of elements

if I have a finite group $G$ and a bunch of elements that are the elements of a set $A$. How can I systematically calculate the smallest normal subgroup of $G$ that contains $A$? I am rather ...
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### Schur Multiplier of general linear group

Ideally I would like to know the Schur multiplier of $Gl(n, F_3)$, but perhaps this is not reasonable to ask. But for a small fixed $n$, this should be known, but i could not find any result when ...
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### Group elements $x$ and $y$ satisfying $x^2 = y^2x^2y$ and $yx^{-1}y^2 = x^7$ commute.

The Question Suppose that $x$ and $y$ are elements of a group such that $$x^2 = y^2x^2y$$ and $$yx^{-1}y^2 = x^7.$$ Show that $x$ and $y$ commute. Motivation This came up in another question, where ...
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In the free group with two generators $F_2\cong\mathbb Z *\mathbb Z$ ($*$ denotes the free product), if two elements $a$ and $b$ commute, then there exists an element $w\in F_2$ such that $\langle ... 2answers 160 views ### Show that$\mathbb{Z}_p\setminus\{\overline{0}\}$is not a group if$p$is not prime. The answer is too short that I think I've gone wrong at some point! Q: If$p$is prime, then the nonzero elements of$\mathbb{Z}_p$form a group of order$p-1$under multiplication. Show that this ... 1answer 79 views ### Density of nilpotent numbers A natural number$n$is nilpotent if every group of order$n$is nilpotent (equivalently, a direct product of Sylow subgroups). A natural number$n$has nilpotent factorization if$\ell\not\equiv1$... 1answer 105 views ### (Theorem) If$G$is a simple group of odd order , then$G \cong \mathbb Z_p$for some prime$p$. I am studying Dumit Foote. I have seen this result in this book. Please help me solve this. Thank you. 1answer 43 views ### A question on the intuition of decomposition of the element of symmetry group Any element of symmetry group$S_{n}$can be decomposed as products of transpositions. Any m-cycle can be decomposed as m-1 transposition products. How should I think of this decomposition? Is there ... 2answers 96 views ### Number of conjugacy classes in finite groups Let$G$be a finite group. Let$C_1,C_2,\dots,C_k$be its conjugacy classes. We denote by$C_{j\ '}=\{g^{-1}|\ g\in C_j\}$the conjugacy class inverse to$C_j$. Set $$a_{rst} = ... 1answer 50 views ### I need an example of a function with these properties. I have a problem that says A is finite and B\subset A and that G is the subset set of S_A consisting of all the permutations f of A s.t. f(x)\in B \ \forall \ x\in B. these functions ... 1answer 66 views ### Action of factor group on a group Suppose A and G are finite groups and A acts on G, written g^a for g\in G, a\in A. If N\unlhd A, does A/N then act on G also? By g^{[a]} = g^a? Or do I need to assume something ... 1answer 49 views ### Group theory problem I am asked to prove "Show that if$${e}<H_1<H_2<...<H_{n-1}<G$$Is a subnormal series for a group G, and if the order of$H_{i+1}/H_i=s_{i+1}$, then G is of order$s_1 ...
I can't solve this problem, please help me! Every group of order 105 is isomorphic to $\mathbb{Z}_5\times H$ where $H$ is a group of order 21.