The study of symmetry: groups, subgroups, homomorphisms, and group actions.

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Number of prime divisors of the order of $E_8(q)$.

I am trying to compute the number of prime divisors of the order of $E_8(q)$. I am interested in the general solution, but in particular, my problem calls for $q=p^{15}$ (for prime $p$) and $q\equiv ...
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382 views

direct product of center of group

Let $Z(G)$ denote the center of a group $G$, let $J_n=Z(G)\times\dots \times Z(G)$, is it true that as a subset of external direct product $G\times\dots\times G$, $J_n$ is a subgroup?normal ...
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121 views

Abstract characterization of $S_5$, why must some involution be in the center of a Sylow subgroup?

I'm trying to follow a sketch proof about the abstract characterization of $S_5$, by Walter Feit. Suppose $G$ is a finite group with exactly two conjugacy classes of involutions, with $u_1$ and ...
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1answer
47 views

Generating a number of a specific order

Here is what I have: select $p$ such that $p - 1$ has a large prime factor $t$: $p - 1 = tu$, where $u$ is a random number $n = p^2 q$, where $q$ is prime pick random $g < n$ and compute $g_p = ...
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293 views

Powers of elements and subgroups

Let $(G,\circ)$ be a group and $N\subseteq G$ a normal subgroup of order $n<\infty$ and let $g\in G$. Is the element $g^n$ in $N$? Given a subgroup $H\subseteq G$ of order $n$, is element $g^n$ in ...
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442 views

Characters of a group with abelian subgroup of index 2

I was reading through a proof this morning which said that the characters of a group with abelian subgroup of index 2 are of degree at most 2. This feels like an easy result but I can't seem to work ...
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84 views

Infinitely many nilpotent elements in $\mathbb{C}[G]$

Suppose $G$ is a finite group and $F$ is a field such that $\mathrm{char}\;F$ doesn't divide $|G|$. Suppose that $F$ is algebraically closed and $G$ is not abelian. How can I prove that $F[G]$ has ...
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123 views

If $(a_1,a_2,…,a_n)\in G_1\oplus G_2 \oplus \cdot \cdot \cdot \oplus G_n$ give a condition for $|(a_1,a_2,…,a_n)| = \infty$

Let $(a_1,a_2,...,a_n)\in G_1\oplus G_2 \oplus \cdot \cdot \cdot \oplus G_n$. Give a necessary and sufficient condition for $|(a_1,a_2,...,a_n)| = \infty$ I know $|(a_1,a_2,...,a_n)|$ is related to ...
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5answers
228 views

How to prove a group has a basis with exactly one element?

I am struggling with the following question. Suppose I have a group $H$ which is a subgroup of $\mathbb{Z}\oplus\mathbb{Z}$, such that any element $\begin{bmatrix} a \\[0.3em] b ...
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146 views

What is the smallest degree of a homogenous polynomial invariant under the action of $D_{2n}$ in the plane.

If we put a regular polygon centered on the origin in $\mathbb{R}^2$ then we can think of $D_{2n}$ as isometries of the plane. What is the degree of the smallest polynomial invariant under these ...
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79 views

Non-identity central element in a central quotient

Let $G$ be a group. Let $Z(G)$ be the center of $G$, the set of elements that commute with every element of $G$. Then, can we say that there is some elements in $Z(G/Z(G))$ which is not $Z(G)$?
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134 views

Why are there $|G/G'|$ 1-dimensional representations of $G$?

Let $G'$ be the derived subgroup of a finite group $G$. We have a correspondence $\{\mathrm{reps \ of \ G/G'}\} \longleftrightarrow \{\mathrm{reps \ of \ G \ with \ kernel \ containing \ G' }\} $ ...
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515 views

The Frattini Subgroup

I am revising for my Group Theory exam and am stuck on the following question; The Frattini subgroup $\Phi(G)$ of a group $G$ is defined to be the intersection of all maximal subgroups of $G$. Prove ...
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164 views

For which of the following values of n is there a group of order n with no proper normal subgroups?

1) n=21 2) n=9 3) n=60 4) n=98 As per the link here http://groupprops.subwiki.org/wiki/Subgroup_of_index_equal_to_least_prime_divisor_of_group_order_is_normal there is least prime divisor of group ...
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214 views

Normalizer and a Sylow subgroup

If $G$ is a group, $P$ is a 5-sylow group of $G$, $N_{G}(P)$ is the normalizer of $P$ in $G$ then: $$[G : N_{G}(P)] \Bigm| [G : P]$$ Why is this true?
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728 views

a free abelian group is torsion-free

let $G$ be a free abelian group of rank $n$ with basis $B=\{b_1,\cdots,b_n\}$ then $G$ must be torsion-free. to prove this let $g=\sum{m_ib_i}\not = 0$ an element of $G$. Suppose there exists $q\in ...
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167 views

Construct group isomorphism $\alpha : \mathbb{Z}^n \rightarrow \mathbb{Z}^n$ s.t. $g=f \circ \alpha$?

Suppose we have surjective group morphisms $$f: \mathbb{Z}^n\rightarrow A \qquad g:\mathbb{Z}^n\rightarrow A.$$ How do I construct a group isomorphism $\alpha:\mathbb{Z}^n \rightarrow \mathbb{Z}^n$ ...
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1answer
180 views

How is the action $(P,Q) \star A=PAQ^{-1}$ decomposed into orbits?

If we let $S$ be the the set of real $m \times n$ matrices and take G to be the direct product $G=GL(m,\mathbb{R})\times GL(n,\mathbb{R})$ and consider the action of G on S as follows $$(P,Q) \star ...
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104 views

If $(\mathbb{Q},+)$ is homomorphic to $( \mathbb{Q}^+, \times)$, then $f(x)=1$ for all $x\in \mathbb{Q}$

If $(\mathbb{Q},+)$ is homomorphic to $( \mathbb{Q^+}, \times)$, then $f(x)=1$ for all $x\in \mathbb{Q}$. This is one of questions from my assignment. I've been working on it for 2 days, but ...
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202 views

A question on Group actions

I was wondering if anyone visiting would be up for solving the following interesting little exercise out of Fulton's Algebraic Topology: A First Course. Let $G$ a group act on a set $Y$. Say that $G$ ...
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82 views

Counterexample of G-Set

If every element of a $G$-set is left fixed by the same element $g$ of $G$, then $g$ must be the identity $e$. I believe this to be true, but the answers say that it's false. Can anyone provide ...
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569 views

How to find the number of homomorphisms of $\mathbb{Z}$ into $\mathbb{Z}$?

Consider the maps $\mu:\mathbb{Z}→\mathbb{Z}$ and $\mu:\mathbb{Z}→\mathbb{Z}_2$. For example if I am asked to find the number of homomorphisms of $\mathbb{Z}$ into $\mathbb{Z}$, and of $\mathbb{Z}$ ...
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449 views

On permutations of left cosets

Let $K$ be a subgroup of some group $H$; let $X$ be the set of left cosets of $K$, i.e. $X = \{hK: h \in H\}$; and let $G$ be the group of permutations of $X$. For all $h \in H$, let $f\,(h) \in G$ ...
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347 views

Prove the determinants of these related matrices are zero.

Suppose you are given this $2 \times 2$ matrix of trig functions: \begin{vmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{vmatrix} The zeros of which give the identity ...
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257 views

Computing Quotient Groups with Infinite Groups

I've asked a similar question: Computing Quotient Groups But now I want to compute a quotient group involving a direct product in which every direct factor is infinite. For example $\mathbb{Z} \times ...
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213 views

Presentation of cyclic group

Let $p$ a prime. Prove that the group $G=\langle x,y: x^{p}=y^{p}=x^{-2}y^{-1}xy=1\rangle$ is cyclic of order $p$.
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335 views

Cosets of the group represented by real numbers excluding zero and multiplication as the binary operation.

I'm having a hard-time understanding cosets when the set in question is uncountable. Here's a simple example that leads me to a contradiction. Let $G = \mathbb{R}\backslash\{0\}$. Then $(G,\cdot)$ ...
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342 views

Is the torsion subgroup the sum or the direct sum of $p$-primary components?

This is written on Page 93 of Derek J.S. Robinson's A Course in the Theory of Groups: Let $G$ be an arbitrary abelian group, $T$ its torsion-subgroup. For an arbitray prime $p$, denote $G_p$ as ...
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648 views

Groups of order 12 without Sylow

It is clear that Sylow theorems are an essential tool for the classification of finite groups. I recently read an article by Marcel Wild, The Groups of Order Sixteen Made Easy, where he gives a ...
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88 views

Non-equivalence of groups and SESs of groups

There is a lemma in Lang's Algebra (2002, page 17): Let $H_1 \triangleleft G_1$, $H_2 \triangleleft G_2$, $f: G_1 \to G_2$, and $H_1 = f^{-1}(H_2)$. Then the homomorphism $$\hat f: G_1/H_1 \to ...
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629 views

every transitive action of an abelian group is regular

Why the following is true : "Every transitive action of an abelian group is regular" Does this mean that every action of an abelian group is free? because as i understand, a regular action is ...
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255 views

Equivalent identification to get the projective plane?

I think $$ \langle a,b | abab= 1 \rangle = \langle a,b | abba = 1 \rangle $$ are 2 equivalent presentations of the fundamental group of the projective plane. To show this, I have tried to transform ...
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114 views

group action in orthogonal decomposition

let $V$ be an inner product space. Let $X$ a subspace of $V$ and $X'$ its orthogonal complement i.e, $V=X\oplus X'$. Let $G$ be a group $G$ acting on $V$. an element in $X\oplus X'$ is ...
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113 views

Definition of identity in a monoid

I'm having trouble understanding the way the identity element is defined in Lang's Algebra. Below is the relevant information. Suppose we have a monoid G with elements $x_{1},...,x_{n}$. We can define ...
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92 views

Product of simple groups has no proper characteristic subgroups?

Let S be a simple group and set $G = S \times ... \times S$. I believe that G has no proper characteristic subgroups but I have no idea about how to prove this. Any help or counterexamples would be ...
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672 views

What are the finite subgroups of $SU_2(C)$?

Is there any reference which classifies the finite subgroups of $SU_2(C)$ up to conjugacy ? What I know is that lifting the finite subgroups of $SO_3(R)$ by the map $SU_2(C) \rightarrow PSU_2(C)$ ...
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198 views

Results for elements in Dihedral Groups: $x \notin \langle r\rangle \Rightarrow rx = xr^{-1}$

As a follow-up to my previous question, the next two exercises state: Use the given generators and relations to show that if $x$ is any element of $D_{2n}$ which is not a power of $r$, then: ...
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156 views

order of a group

let $D = <x, y | x^2, y^2, (xy)^n>$. What is the order of $D$? Thank you very much.
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1answer
168 views

Embedding of general linear groups

Can we embedd $GL(2n,q)$ into $GL(n,q^2)$ for $n\in \mathbb{N}$ and $q=p^m$, $p$ a prime? If yes, how?
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362 views

Question about direct sum of groups

I have trouble understanding this question and its supposed answer. Let $A$ be a subgroup of $G$. Prove that $A$ is a direct summand of $G$ if and only if there is a homomorphism $p:G \to A$ with ...
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144 views

Simplified Version: $N \unlhd F$, $a,b \in N$. $[F:N]=p$, prime. $a \sim b$ in $F$ (conjugates). Are $a \sim b$ in $N$? (more inside)

I'm trying to understand the proof for Lemma 4.8 in Lyndon & Schupp Combinatorial Group Theory, page 26, proposition 4.8. In the very end of this proof, we have: $N \unlhd F$ where $F$ is a ...
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314 views

Are they isomorphic?

$G$ and $G \times G$ where $G = \Bbb Z_2 \times \Bbb Z_2 \times \Bbb Z_2 \times\cdots$ The answer says yes but I cannot figure out what homomorphism function I could use.
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147 views

Relationship between topology of a compact group and the topology of its profinite completion

Suppose $G$ is a compact topological group. We can construct the profinite completion of $G$; let's call this $\Gamma$. My questions are: 1) Assuming that we know nothing about the (original) ...
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24 views

For a $p$-group $G$, and $H \le G$, if $G=H G^\prime$, then $H=G$ [on hold]

For a $p$-group $G$, and $H \le G$, prove that, if $G=H G^\prime$, then $H=G$; where $G^\prime$ is the commutator subgroup of $G$.
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42 views

Cayley graph is a tree iff group is free

I am looking at this proof of this claim that the cayley graph is a tree iff g is a free group with generating set S. For the direction '$\implies $' I see that they have assumed that there are two ...
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53 views

Pushout of a subgroup

Let $G$ be a group, $H\subseteq G$ be a subgroup and $\iota: H\to G$ the inclusion map. What is the "pushout along $\iota$? How is it constructed?
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35 views

Is $\mathbb{Z}\times\mathbb{Z}/((6,5),(3,4))$ is finitely generated?

Let $A$ be the quotient of the free abelian group $\mathbb{Z}^2$ by the subgroup generated by $(6,5)$ and $(3,4)$. the question is $A$ is finitely generated? and if Yes. Can we Decompose it into a ...
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36 views

About ring $( \frac{F[t]}{t^2})$

I was trying to study some properties of the group $GL_2 ( \frac{F[t]}{t^2})$ where $F$ is a field with $p$ elements. $p$ is a prime. I found that $ \frac{F[t]}{t^2}$ is a discrete Valuation ring. ...
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49 views

If $\left<3\right> \cong \mathbb Z_2$ and $\left<2\right> \cong \mathbb Z_3$, why isn't $\left<3\right>\left<2\right> \cong \mathbb Z_2\mathbb Z_3$

$\left|\mathbb Z_{6}\right|=6=2 \cdot 3$. Since $\mathbb Z_{6}$ is abelian, all subgroups are normal and thus its Sylow subgroups are unique. $\left<3\right>$ is the $2$-Sylow, and ...
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46 views

Question about relationships between images and kernels of a group

I have been having a reoccuring problem in my abstract algebra class with my professor defining notation and using things different from the books for homework and it's giving me difficulty to follow. ...