# 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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### Question on Subgroups and Left Cosets

Here is an unanswered exercise from class notes. For the group <{$i,-i,1,-1$},*> and a subgroup H where $|H|$ = 2. Find the left cosets of G induced by H. G={$i,-i,1,-1$}
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### Subgroups of Sufficiently Large Symmetric Groups / Cayley's Theorem explanation

Here's the question: is every finite group a subgroup of a symmetric group of sufficiently large order? More specifically, if a group $G$ has order $n$, then is it true that $G \le S_{n}$? For ...
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### The order of the representative elements of conjugacy classes

Suppose $A$ is an arbitrary subset of group $G$ and $K_G(A)$ be the number of $G$-conjugacy classes contained in $A$. A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ ...
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### groups generated by two elements of order 3

I'd like to know whether it is possible to find a characterization (cardinality?) of the set of finite, non-abelian groups generated by two elements $a$ and $b$ whose order is $3$? Is it the same task ...
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### representation of extraspecial group

Let $H=\langle x,y\rangle$ be a non abelian group of order $p^3$ which has exponent equal to $p$. Also let $G$ be an extraspecial group of order $p^{2r+1}$ exponent $p$. We know that $G$ is isomorphic ...
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### How can we find element orders of a finite group?

Suppose $p$ and $q$ are two prime divisor of the order of a finite group $G$. I want to know if $G$ has an element of order $pq$ using the character table of $G$. Is this possible? If so, please ...
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### Do these definitions make sense?

Letting $G$ be a group and $S(G)$ be all permutations of $G$, define $$L(G)=\{\phi\in S(G)|\forall n\in \mathbb Z , \phi(g^n)=\phi(g)^n\ \ \forall g\in G \}.$$ It is easy to check that $L(G)$ is a ...
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### What are all the automorphisms of a group of order $9$ generated by two elements?

Let $G$ be a group of order $9$ generated by two elements $a$ and $b$ such that $a^3 = b^3 =e$. How to determine all possible automorphisms of $G$?
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### How is this subgroup abelian?

Let $G$ be a finite group of order $2n$ such that half of the elements of $G$ are of order $2$ and the other half form a subgroup $H$ of order $n$. Then I know that $H$ is of odd order because for ...
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### Fixed point set of orthogonal Transformation

I need some help with this problem. Let $g$ be an element of the orthogonal group and $s$ a reflection. Then the dimension of the fixed point set of $g$ and $gs$ differ by $\pm 1$. Since that ...
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### groups of order $p^5$ and exponent p

We know that a group of order $p^5$ is an extra special group. How we can show it doesn't have any abelian subgroup of order $p^4$? Also what is the presentation of this group if its exponent is equal ...
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### Generating a subgroup

While getting ready for my exams I am trying to solve this question "Find a Sylow $2$-subgroup in $S_4$ which contains the element $(1, 3, 2, 4)$." I started by permutation of the element required ...
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### Is a finite group with a certain automorphism must be abelian

Let $G$ be a finite group, and let $f:G \rightarrow G$ be an automorphism, such that $x f(x) f(f(x))=1$ for any $x \in G$. Is $G$ must be abelian? I believe that there are examples where $G$ is not ...
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### Hopefully easy Lang-Steinberg computation: for Weyl elements

How do you write an element of the Weyl group as $g^{-1} F(g)$? For instance, let $G = \langle x_1(t), x_2(t), x_{-1}(t), x_{-2}(t) : t \in K \rangle$ where $K$ is an algebraically closed field ...
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### Permutations: If I know $\alpha$ and the cycle structure of $\alpha\beta$, can I find $\gamma$ for which $\gamma\beta$ also has this cycle structure?

Suppose we have two permutations $\alpha$ and $\beta$ (of a set $S$ of size $|S|=n$), and I know $\alpha$ and the cycle structure of $\alpha\beta$. But I don't know $\beta$. Can I find a ...
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### Operations in finite field $F_p$

If I have a finite field $F_p$, where $p$ is prime how can I define operations like $+, -, \times, /$? Can I just make: $$add: (a + b) \mod p$$ $$sub: (a-b) \mod p$$ $$mul: (a\times b) \mod p$$ ...
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### Has a finite group, generated by $a,b$ always a relation of the form $1=a b a^{\alpha} b^{\beta}$?

Question as stated in the title: Has a finite group, generated by $a,b$ always a relation of the form $1=a b a^{\alpha} b^{\beta}$? If not, can you give me a counterexample? Thanks
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### Show that $|Aut(G)|<n^{\log_2(n)}$ where $G$ is finite

Let $G$ be a finite group of order $n>1$. Show that $|Aut($G$)|<n^{\log_2(n)}$. The 'only' thing I know is the group of inner automorphisms is isomorphic to $G/Z(G)$ and by definition, ...
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### Determining a group $G$ by looking at the number of homomorphisms $H\to G$

I read somewhere that, given a finite group $G$, its structure is completely determined from the knowledge of the values of $|\{H\to G\}|$ (the number of homomorphisms from $H$ to $G$) as $H$ varies ...
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### Isomorphism of Groups (C.S.I.R)

Suppose $(F , +, .)$ is the finite Field with $9$ elements. Let $G = (F , +)$ and $H = (F \setminus \{0\}, .)$ denotes the underlying additive and multiplicative groups respectively. Then which are ...
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### Index and normal subgroups

I want to show the following. For an infinite group G with only two normal subgroups (G and {e}) holds: There does not exist a non-trivial subgroup of G with finite index. I think i should prove ...
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### Simple groups of order 7920

I would like to prove that every finite simple group of order 7920 is isomorphic to $M_{11}$. I would be appreciated if someone can give me some points.
### Mapping from $\{1,\ldots,n!\}$ to the symmetric group $S_n$
Is there an easy known bijective mapping formula between the set $\{1,\ldots,n!\}$ and the symmetric group $S_n$? I want to pick a number $k \in \{1,\ldots ,n!\}$ and assign a unique permutation of ...
Let $G$ be a group (say, finite) and let it act on a set $X$ (say, also finite). For every element $g \in G$, we can consider its action on $X$. My rather vague question is What information about ...