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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$\mathbb{Q}(\zeta_p)$ contains only one subfield $L$ such that $[L : \mathbb{Q}]=3$

Suppose $p$ is a prime number, $p\equiv1$ mod $3$ and $\mathbb{Q}(\zeta_p)$ is the $p$-th cyclotomic extension. Prove that $\mathbb{Q}(\zeta_p)$ contains only one subfield $L$ such that $[L : ...
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To prove that a hopfian group is not free.

In the book "presentation of groups by johnson" in page $36$ they are trying to obtain a group that is locally free but not free (Example $2$).They have proved that the group $U=\bigcup_{n \in ...
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Normal subgroups of matrices

Let $G=\begin{bmatrix}1&a\\0&b\end{bmatrix}$ so that $a,b\in\mathbb C$ and $b\ne0$. I need to prove that $G$ has infinitely many normal subgroups. I attempt to do this by constructing ...
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374 views

What's the difference between Abstract Algebra and Group Theory?

I'm slowly beginning a student of certain higher mathematics. I'm trying to see if I would prefer to study Group Theory or Abstract Algebra. I know that Abstract Algebra seems to "come before" ...
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13 views

Generator of $Gal(K/\mathbb{Q})$

Let $K=\mathbb{Q}(\zeta_5)$. Prove that there is a $\tau \in G$ such that $\tau \zeta_5=\zeta_5^2$ is a generator of $Gal(K/\mathbb{Q})$ I belive we must consider $\mathbb{Z_5}$, but I am not ...
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Minimal polynomial using Galois theory

I have a couple of questions, given below, about the following problem from a course in Galois Theory. Let $K=\mathbb{Q}(\zeta_{13})$. $K$ contains a unique subfield $L_4$ such that $[L_4 : ...
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84 views

$(C_2 )^3$ is not a subgroup of $S_4$

Prove $(C_2)^3$ is not a subgroup of $S_4$. (Using group actions.) I could think of a permutation argument that $(C_2)^3$ is not a subgroup of $S_4$. But I would like to argue it by considering ...
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Galois subfields and subgroups

Find the minimal Galois extension $L$ of $\mathbb{Q}$ containing $\mathbb{Q}(\sqrt[4]{5})$ $L=\mathbb{Q}(\sqrt[4]{5}, i)$ is a splitting field of $X^4-5$ over $\mathbb{Q}$ Describe the ...
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50 views

Determing whether a subgroup is normal

I have been working with normal subgroups and feel like I am doing something wrong. I understand there are many ways to demonstrate if a subgroup is normal, but the methods seem to take longer than I ...
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20 views

If a normal subgroup shares elements with a conjugacy class, then it contains it entirely?

One of my group theory review problems seems to follow directly from definitions, but I'm not sure. The problem is: Let $G$ be a group and $C$ a conjugacy class of $G$. Let $N$ be a normal subgroup ...
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Put $C_{12}\times C_{35} \times C_{45}$ is canonical product

$C_{12} \cong C_3 \times C_4$ $C_{35}\cong C_5\times C_7$ $C_{45} \cong C_5 \times C_9$ Then you combine these and rearrange into factors according to the prime involved but there is no prime that ...
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55 views

What does it mean for an automorphism to centralize factor group $G/M$?

Let $G$ be a group and $M$ be a normal subgroup of it. An automorphism $\phi$ centralizes the factor group $G/M$. What does it mean for an automorphism to centralize a factor group $G/M$? I ...
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15 views

Showing that a subnormal series for a finite group $G$ can be made into a composition series for $G$

Suppose that $\{e\} < G_1 < G_2 < \cdots < G_n = G$ is a subnormal series, thus for all $i$, we have $G_i$ is normal in $G_{i+1}$. How can I show that this can be "refined" to a ...
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39 views

Finitely generated vs infinitely generated group

When we say a group $G$ is finitely generated we mean it can be generated by a finite number of elements but this does not exclude the possibility of being generated by an infinite number of elements ...
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Compute the Galois group of $p(x)=x^4+ax^3+bx^2+cx+d$

Compute the Galois group of the following polynomial: $$p(x)=x^4+ax^3+bx^2+cx+d$$ Step 1: Calculate cubic resolvent Step 2: Calculate the discrimimant $\gamma$ of the cubic resolvent Step 3: ...
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Quasi-Group represented by a graph which is not a Triangle-Free Graph locally

Can each of all quasi-groups be represented by a graph (latin square graph), which is not locally triangle free graph ? Quasi Group can be represented by Latin Square matrix, thus by a Latin ...
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75 views

Can a triangle free graph represent a Group?

Some facts are- Group can be represented by a graph. Group Isomorphism $\leq_p$ Graph Isomorphism. Under this context, my questions is- Can a triangle free graph represent a group? Edit: My ...
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Showing that a quotient group $G/N$ is isomorphic to $\mathbb{Z}_3$

I have permutations $\sigma=(135)(27)$, and $\tau = (27)(468)$. $G =\langle \sigma,\tau \rangle$ and $N$ is the smallest subgroup of $G$ that contains $\tau$, so $N = \langle \tau \rangle$. $|\sigma| ...
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23 views

If a subset of a free group $F$ is Nielsen reduced, then it is a basis of $F$. Is the converse statement true? [on hold]

If a subset of a free group $F$ is Nielsen reduced, then it is a basis of $F$. Is the converse statement true? I mean if I take a basis $U$ of $F$, then is it true that it has to be Nielsen reduced? ...
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A surjective homomorphism and normal subgroups [duplicate]

I'm reviewing group theory for a comprehensive exam and this question came up. Suppose I have two groups $G$ and $K$ and $\varphi$, a surjective homomorphism from $G$ to $K$. How can I prove that ...
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21 views

Expressing a permuation as a product of disjoint cycles.

The theorem: Let $p$ be a permutation of $\{1,\ldots,n\}$. Then $p$ can be expressed as a product of disjoint cycles. How would you express a permutation that permutes every element of the set, ...
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Difference between symmetry algebra and symmetry group

What is the difference between symmetry algebra and a symmetry group? I just wanted to know if my understanding is right. Lets say we have a system of differential equations. Then the symmetry group ...
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Find the order of $U_{2n}$

Let $n$ be an odd integer and let $k$ be the number of elements in $U_n.$ What is the order of $U_{2n}$? I have said $\left\lvert U_{2n}\right\rvert=\varphi(2n)=\varphi(2)\varphi(n)=\varphi(n)=k.$ ...
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Why all irreducible representations appeear in the regular representation?

Let $G$ be a finite group and $R$ the regular representation. That is, as a vector space $R = F(G)$ is the free vector space with basis $G$. If the basis is $\{e_g : g \in G\}$ the action is defined ...
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20 views

Local group rings

Let $k$ be a field of characteristic $p$ and $G$ a finite group. How do you prove that if $kG$ is local then $G$ is a $p$-group? (I know how to prove the converse but not this implication).
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187 views

Bound on the size of Permutation Set for Isomorphism

$\textbf{Claim :}$ $G, H $ are partitioned into sub-graphs $\{ G_1,G_2 \cdots G_x \}$ and $\{ H_1,H_2 \cdots H_x \} $ . For each $G_i$ we constructed a set permutation, $\beta_i$ such ...
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Isomorphisms concerning group of units

How is it possible to show that $U_2^k \cong \mathbb{Z}_2 \times \mathbb{Z}_2^ {k−2}$ for $k\geq 3$?
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Showing a consequence of definition of internal direct products.

Show that if $G$ is the internal direct product of $H_1,H_2,\dots ,H_n$ and $i\neq j$ with $1\leq i\leq n,1\leq j\leq n$, then $H_i\cap H_j=\{e\}$. The definition that I follow is as follows: ...
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27 views

Group extension that doesn't realize a coupling

Let $E$ be an extension of $N$ by $G$: $$N \hookrightarrow E \twoheadrightarrow G$$ If $N$ is abelian, then $E$ uniquely defines an action of $G$ on $N$. More generally, it defines a unique class ...
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Examples of nilpotent connected locally compact groups which are not Lie groups

I am looking for examples of nilpotent connected (or at least almost connected) locally compact groups which are not Lie groups. Do you know of such examples ?
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a conjugate of a glide reflection by any isometry of the plane is again a glide reflection

Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length. I know that: the conjugate ...
3
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1answer
43 views

Is there any conclusion about a group, if the group has unique element of order $n>1$?

If a group $G$ has an unique element of order $n>1$, then which of the following is true: Order of $G$ is $n$. Order of $Z(G)$ is greater than $n$. $Z(G)=G$ $G=S_2$ (I've seen that (1) can not ...
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Subgroups of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$

If I'm understanding the main theorem of (infinite) Galois theory correctly, applied to $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, it gives us: a) all its open subgroups are ...
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24 views

If the correspodence $aHbH=abH$ defines a group operation on the set of left cosets of $H$ in $G$, then show that $H$ is normal in $G$.

If the correspodence $aHbH=abH$ defines a group operation on the set of left cosets of $H$ in $G$, then show that $H$ is normal in $G$. My attempt: Let $x\in G$. Then we know that $xHx^{-1}H=H$. ...
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$S_4/H \simeq S_3$ where $H$ is a normal subgroup

Prove that the group of permutations of four symbols $S_4$ contains a normal subgroup H such that the quotient group $S_4/H$ is isomorphic to the group of permutations of three symbols $S_3$. ...
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Proof of unicity of decomposition of a representation

I'm studying representation theory and in the book the author makes the following proposition with the following proof: Proposition: For any representation $V$ of a finite group $G$, there is a ...
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Show that the permutation $(1 \space 2 \space 3)$ can not be a cube of any element of $S_n.$

Here is my try: If there exists $a \in S_n$ such that $a^3=(1 \space 2 \space 3)$, then $a^9=e$ where $e$ is identity in $S_n$. Then $o(a)=9$. I don't know how to proceed further. Can anyone ...
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Finite abelian group $H$ and $a\mid \exp(H)$, then $H$ has an element of order $a$

Can't find any theorem or helpful ideas that might link to this. In all honestly, I am very lost in this topic. If $H$ is finite abelian group and some $a$ such that $a\mid\exp(H)$, then $H$ has ...
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1answer
25 views

Describe all extension groups of a given subgroup $H \trianglelefteq$ Aff$\mathbb{(F_q)}$ by Aff$\mathbb{(F_q)}/H$

Let $\mathbb{F_q}$ be a finite field. Consider the group Aff$\mathbb{(F_q)}$ Aff$\mathbb{(F_q)} := $ $ \ \begin{Bmatrix} \begin{pmatrix} a&b\\ 0&1\\ \end{pmatrix} \colon a, b \in ...
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406 views

How many different necklaces can be make from 8 blue beads, 3 green beads, and 3 brown beads?

I am trying to figure out how many different necklaces can be make from 8 blue beads, 3 green beads, and 3 brown beads. I understand how to do the problem with two colors, but I am struggling to ...
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Problems in understanding a passage in the proof of Grün theorem for transfer

This is the statement of the theorem: Let $P$ a Sylow $p$-subgroup of $G$ and $Z$ a subgroup of $Z(P)$ that is weakly closed in $P$. Set $H=N_G(Z)$. Then $P\cap G'=P\cap H'$ and $P/(P\cap G')\simeq ...
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Finding centralizer of a matrix in general linear group.

I saw the following question from Gallian's book on abstract algebra. I am required to find the centralizer of the matrix $$A= \begin{bmatrix} 1 & 1 \\ 1 & 0 \\ ...
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1answer
34 views

Is the homomorphism $f: n\mapsto e^{in\theta}$ injective?

Would any of you mind just taking a look to see whether this is a valid proof? I'm trying to prove whether the homomorphism $f: \mathbb{Z} \rightarrow U(1)$ defined by $f(n) = e^{i n \theta}$ is an ...
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Does there exists an additive group homomorphism between two $K$-vector space that is not $K$-linear

My question is: Give me a field $K$. Can we always find two $K$-vector space $V_{1}$, $V_{2}$ and a map $f:V_{1}\rightarrow V_{2}$ such that: (1) If we view $V_{1}$, $V_{2}$ as additive group, then ...
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Prove that $[G\times H : A\times B]=[G:A][H:B]$ when $A < G$ and $B < H$.

The original question is that: If $A$ is subgroup of group $G$ and $B$ is a subgroup of group $H$, then express $[G\times H : A\times B]$ in terms of $[G:A]$ and $[H:B]$ and prove the result is ...
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53 views

Show that $x^{-1}$ has finite order $n$

Let $G$ be a group and let $x\in G$ have finite order $n$. Show that $x^{-1}$ also has order $n$. I'm comfortable with showing that $$(x^{-1})^n=(x^n)^{-1}=e^{-1}=e$$ However, I don't feel that ...
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35 views

Polynormal subgroup

Let $G$ be a group. $H$ is said to be polynormal in $G$ if for each $x\in G$, we have $H^{\langle x \rangle} = H^{H^{\langle x \rangle}}$ where $H^{\langle x \rangle} = \langle x^nHx^{-n} \;|\; n\in ...
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Kernels in $\mathbf{Top}$

There is a following well-known theorem for abelian categories (at least the ones I know, Ab, $R$-mod and so on... not so familiar with categorical language to be honest) which states the following : ...
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Construction of a specific non-commutative and infinite group (with conditions on the order of the elements)

I am struggling with the following problem: Find a group $G$ such that whenever $m, n, k \geq 2$ are natural numbers, then there exist $a, b \in G$ such that the order of $a$ is $m$, order of $b$ ...
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495 views

Subgroup structure of dihedral groups

I am currently looking into structure of dihedral groups; I am interested in their subgroup structure. Dihedral group have two kinds of elements; I will use their geometric meaning and call them ...