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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Sylow theorem Cyclic sylow p subgroup

$p$ is the smallest prime dividing order of $G$. $P$ is a sylow p subgroup which is cyclic. Prove that $N_G(P) = C_G(P)$ This is my approach : Since $P$ is sylow p subgroup so its order is some power ...
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48 views

Subgroup of a (free) group.

Let $F$ be a group, generated by $x_1,...,x_m$ and $H$ be its subgroup such that $|F:H|=n < \infty$. How to prove that $H$ can be generated by $n(m-1)+1$ elements?
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90 views

Decomposition of representation of symmetric group

Let $V$, $\dim V=n-1$ be the standard representation of the symmetric group $S_n$ and let $V'= \langle x_1,x_2,\ldots,x_n \rangle$ be its natural representation. Then ( see. Fulton, Harris, 4.19) ...
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49 views

Each irreducible representation is a subrepresentation of induced one

I'm learning what irreducible representation is and need some examples. One of them is as follows: Let $G$ be any group and $H$ - it abelian subgroup. How to prove that each irreducible representation ...
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41 views

Irreducible representations and abeliean subgroups

There is a theorem in representation theory which is surprising to me: the dimension of irreducible (complex) representation of finite group is not greater that $(G:H)$ - an index of abelian subgroup ...
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46 views

Prove that $\varphi$ is automorphism

$G$ is commutative group. $|G|=n$. $m\in \mathbb{N}$ and $\gcd(m,n)=1$. I need to prove that $\varphi :G\to G$, $\varphi(x)=x^m$ is automorphism of G. My try: I assume that $a\in \ker(G)$, so $a\in ...
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64 views

If $|G|=21$, show that at least one of the Sylow $p$ subgroups is normal.

I have done something but I am not sure if it's correct. So $|G|=21$, $21=7\cdot3$ $p=7 \rightarrow n_7=\{\text{factors of }3\}=\{1,3\}\equiv 1 \mod 7$ $3$ isn't congruent $1\pmod 7$. $1$ is ...
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47 views

What is the value of $\sum_{i=1}^n X_i(g) X_i(h^{-1})$ when $g,h \in G $ are in the same conjugacy class?

I know the value of this summation $\sum_{i=1}^n X_i(g) X_i(h^{-1})$ when $g,h \in G $ are in the different conjugacy class will be zero and I know how to prove it but what about if they are in the ...
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21 views

Construction of an isomorphism between certain subgroups of $GL_2(\mathbb{C})$ and $S_4.$

Consider the following matrices $A:=$$ \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} ,\ B: =\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ of multiplicative group $GL_2(\mathbb{C})$ and ...
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93 views

Alternating group $\operatorname{Alt}(n)$, $n>4$ has no subgroup of index less than $n$.

I have read somewhere (do know where) the following statement. Alternating group $\operatorname{Alt}(n)$, $n>4$ has no subgroup of index less than $n$. I want to prove it. If there is a ...
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84 views

Orders of Elements in Minimal Generating sets of Abelian p-Groups

I'm looking for as much information about the orders of elements in minimal generating sets of finite abelian $p$-groups as possible. What I really need is complete knowledge about the possible orders ...
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49 views

Product of three supersoluble subgroups

I need a proof of the following statement: Let a finite group $G=AB=AC=BC$ be the product of three supersoluble groups $A$, $B$ and $C$. If the commutator subgroup $G'$ is nilpotent, then $G$ is ...
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59 views

Normal and subnormal series

Let $G$ be a subnormal series of a finite group $G$ then is there any specific formula to find the order of $G$? Thanks for advance.
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75 views

Lift a group action from a quotient

Let $p$ be a rational prime and $H$ be a finite cyclic group of prime order $l$ prime to $p$, i.e. $(l,p) = 1$. Let $G$ be a finite abelian group of $p$-power order. If I can write an (abelian) group ...
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52 views

Index of centralizer of alternating group

For any element $x \in A_5$, we have that $$[A_5:C_{A_5}(x)]=\begin{cases} [S_5:C_{S_5}(x)], & \text{condition 1} \\ \frac{1}{2}[S_5:C_{S_5}(x)], & \text{condition 2} \end{cases}$$ Basically, ...
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Partial converse to fact about isomorphic finite groups and their representations

If two finite groups are isomorphic then, they have the same irreducible characters (if $G_1\cong G_2$, we must send elements in a conjugacy class of $G_1$ to elements of the corresponding class in ...
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108 views

Lower bound for the order of a group element

I would like to know how to find a lower bound of an element in a large group. Let's say I have an element $x\in G$ and that $|G|=N$ is very large, say $\mathcal{O}(10^{300})$. To find the order of ...
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45 views

correspondence theorem question

If G has order 12 and G' has order 6, produced by elements x and y respectively, and phi maps G to G' which is defined by phi(x^n)=y^n, how is the correspondence exhibited in the correspondence ...
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What are the cosets of this presentation?

I'm reading a book on algebra, and they give a presentation for $S_3$, with 6 elements $\{1, x, x^2, y, x y, x^2y\}$ as $$x^3 = 1,\quad y^2 = 1,\quad y x=x^2y$$ Now later in the book, there is a ...
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54 views

Upper central series in Coclass Theory.

It is proved by Aner Shalev, that for any finite $p$-group of coclass $r$(and sufficiently large order), there is some severe restrictions on lower central series $(\gamma_i(G))$. For instance, ...
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54 views

On some endomorphisms of finite groups of odd order.

Let $G$ be a group of odd order. It is known that if every central automorphism of $G$ acts trivially on the center, then $G$ is purely non-abelian, this amounts to saying that every central ...
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81 views

Structure of finite abelian group

I am trying to solve this problem, and I think I should use the structure theorem for finite abelian groups, but i can't really figure this out. Let $G$ be a finite abelian group. Prove there is a ...
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101 views

Two questions in Isaacs' book Finite Group Theory

I am reading Isaacs' book finte group theory, and I have two questions. in page 90, there is a Wielandt's theorem (if $G$ has a nilpotent Hall $\pi$-subgroup, then all Hall $\pi$-subgroups of $G$ ...
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86 views

The conjugacy classes of the simple group PSL(2,q)

If $q=p^{\alpha}$, where $p$ is a prime number, then I would like to know the number of conjugacy classes related to elements of order $p$ and $2$ in the simple group $PSL(2,q)$.
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148 views

Cayley theorem to prove that a group of order $2^mk$, $k$ odd, can't be simple

I have to solve this exercise WITHOUT Sylow theorems and Cauchy Lemma. In fact this exercise is given in the Cayley's theorem section. let $G$ be a group of order $2^mk$, where $k$ is odd. Prove ...
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Simple finite supergroups and sporadic supergroups

Is there an analogue list with "finite simple supergroups" similar to the finite simple group classification? Are there sporadic "finite simple" supergroups?
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63 views

Finite $\Delta$-module of $p$-power order

I have a question concerning lemma, that I want to prove: Let $p$ be a prime and $\Delta$ be a finite group of order prime to $p$. Let $M$ be a finite $\Delta$-module of order a power of $p$. Then ...
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48 views

A group of order 2520

Let $G$ be a group of order 2520 and let $K$ be the maximal normal soluble subgroup of $G$. If we know that $G/K\cong A_5$, $K=C_2\times (C_7 :C_3)$, $C_G(K)=SL(2,5)$ and $G= C_G(K) K$, what would be ...
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Values of virtual characters

Let $G$ be a finite group. Let $K$ be a number field and $K^c\subset\mathbb{C}$ its algebraic (separable) closure. Denote by $R_G$ the additive group of functions generated by the characters of ...
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$G$ be a non-nilpotent and supersoluble finite group.

Let $G$ be a non-nilpotent and supersoluble finite group. $G=Q\ltimes P$ and $|Q: C_{Q}(P)|=q$ is prime, Where $P$ is a group of prime order $p\neq q$. $Q$ is noncyclic group of order ...
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$G$ be a $p$-supersoluble group. and $O_{p'}(G ) = 1$ then $G$ is supersoluble

Let $G$ be a $p$-supersoluble group. If $O_{p'}(G ) = 1$ then $G$ is supersoluble.
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Non-abelian groups of order $p^2q$

Let $G$ be a non-abelian group of order $p^2q$ and $p> q$. i) $p \equiv 1 \mod q$ or $p \equiv -1 \mod q$; ii) The number of elements of order $pq$ in group $G$ is $0$ or $p(p-1)(q-1)$.
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Assume simply connectivity without loss of generality

Let $X$ a connected Riemann surface and $G$ a finite group that acts faithfully and holomorphically on $X$. Further, let $x \in X$ a non-trivially stabilized point (we know these points are discrete), ...
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50 views

How to build $\operatorname{Hom}(\mathbb{Z}_p^*,\mathbb{Z}_{pq}^*)$ without solving DLP?

For given two distinct primes p and q, I want to construct all homomorphisms from the multiplicative group $\mathbb{Z}_p^*$ to the multiplicative group $\mathbb{Z}_{pq}^*$. Thanks to Jyrki ...
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Supersolvable group and nilpotent maximal subgroup

$G$ is a supersolvable group, $|G|=pq^{b}$ ($p$ and $q$ are different prime numbers). $Q$ is a $q$-Sylow subgroup of $G$, $Q=\langle x\rangle$ and $\operatorname{Cor}_{G}(Q)=\langle x^{q}\rangle$. If ...
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44 views

Maximal subgroups of $G=Z_{3}\ltimes Q_{8}$

Let $G=Z_{3}\ltimes Q_{8}$. How can find Maximal subgroups of $G$ ? $$Q_{8}$$ is Quaternion group of order of 8 and $$Z_{3}$$ is cyclic group of order 3
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50 views

A family of $8$-regular Ramanujan Cayley Graphs

I'm looking for expander graphs with certain properties. Is there a family of $8$-regular Ramanujan Cayley graphs $\{\text{Cay}(G_n,S_n)\}_n$ such that each of them has no cycles of odd length? ...
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68 views

Constructing automorphisms

Given the nonabelian group $G$ of order $p^3$ where $p$ is a prime satisfying $p\equiv1\pmod4$, that is $G=\mathbb{Z}_p^2\rtimes\mathbb{Z}_p$ or $\mathbb{Z}_{p^2}\rtimes\mathbb{Z}_p$. Let $H$ be a ...
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92 views

Generators in $p$-groups

Let $G$ be a non-abelian finite $p$-group such that $G/[G,G]$ is elementary abelian group of order at least $p^3$. Let $S=\{\bar{x_1},\bar{x_2},\bar{x}_3,\cdots,\bar{x}_n\}$ be independent generators ...
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125 views

Order of kernel of a homomorphism

Let $p,q$ be distinct primes. Prove that the kernel of the map $$f: (\mathbb{Z}/p^k\mathbb{Z})^* \rightarrow (\mathbb{Z}/p^k\mathbb{Z})^*$$ defined by $f(x)=x^q$ has order $\gcd(p-1,q).$ Thank ...
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Efficiency of Group-Theoretic Algorithms in MAGMA

Given a finite permutation group $G$ and an element $a\in G$ with conjugacy class $X$, I am interested in determining when for a given element $x\in X$ the subgroup $<a,x>$ generated by $a$ and ...
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elementwise conjugate but not conjugate homomorphisms

Does there exist a finite group $G$ and two group homomorphisms $\rho_1,\rho_2:G\to PGL(2,\mathbb{C})$ such that (i) For all $g\in G$ there exists $M=M(g)\in PGL(2,\mathbb{C})$ such that ...
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An example of a group

I need an example of a finite group $G$ by the following properties: 1) Order $G$ is $336$. 2) For every prime $p$, $G$ has not any elements of $7p$. 3) the number of Sylow $7$-subgroups $G$ is ...
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Question on restriction of irreducible group representations to normal subgroups

I am confused about the answers to the following question: Restriction to a normal subgroup with the original question copied here: Let $A$ be a normal subgroup of a finite group $G$ and $V$ an ...
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140 views

Regular subgroups of affine general linear groups

Let $AGL(2d,2)$ be the affine general linear group acting natrually on a $2d$-dimensional vector space over $GF(2)$. Is there a regular subgroup of $AGL(2d,2)$ isomorphic to $Z_{2^d}:Z_{2^d}$ for ...
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Product of subgroups and generating sets

Prove or disprove the following: $(1)$ Let $H$,$K$ be subgroups of a finite group $G$. If $|HK|=|\langle H,K \rangle|$, then $HK=\langle H,K \rangle$, that is, $HK$ is a subgroup of $G$. $(2)$ Let ...
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107 views

Hall subgroup of a finite group?

How did the author get that $L=(L \cap H)(L\cap K)$ in Lemma $5$ below. Remark: All the groups here are finite. $H$ permutes (commutes) with $K$ means $HK=KH$ where $H$ and $K$ are subgroups of some ...
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64 views

Soluble subgroups of finite classical simple groups

What do we know about the soluble subgroups of finite classical simple groups (and maybe their automorphism groups)? For example, are their maximal soluble subgroups explicitly known?
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27 views

Degrees of parabolic subgroups

Suppose a finite reflection group $G$ has the degrees $d_1,\ldots,d_n$. Let $G^*$ be a parabolic subgroup of $G$. What are the degrees of $G^*$. Since $|G^*|$ divides $|G|$ it is clear that the ...
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What are various approaches of proving that a permutation group is a subgroup of another?

I have two permutation groups $G_1=\langle g_1,g_2\rangle$ and $G_2=\langle h_1,h_2\rangle$ acting on $\Gamma_1$ and $\Gamma_2$ respectively. I want to prove that $G_1 \leq G_2$. What are more ...