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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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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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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3answers
36 views

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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12 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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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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34 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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33 views

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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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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66 views

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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1answer
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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13 views

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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20 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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22 views

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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1answer
39 views

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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19 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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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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24 views

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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1answer
46 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 ...
2
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1answer
26 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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54 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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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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17 views

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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2answers
38 views

$S_4/H \simeq S_3$ 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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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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2answers
80 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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1answer
33 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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1answer
22 views

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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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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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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1answer
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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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1answer
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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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Show that the product of two transpositions can be expressed as a product of $3$-cycles

Consider the symmetric group $S_n$ where $n>2.$ Show that the product of two transpositions $(ab),\,(cd)$ can be written as a product of $3$-cycles where $a,b,c,d$ are all distinct. I'm not sure ...
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When this map $f(x)=x^2$ will be a automorphism on $G$, where $G$ is a commutative group.

I know that for a commutative group $G$ the map $f(x)=x^2$ is a homomorphism from $G$ to $G$. My question : When this map $f$ will be a automorphism on $G$, where $G$ is a commutative group. In other ...
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If $a^3=1$, is $G$ abelian?

If $G$ is a group that satisfies $a^3=1$ for every $a\in G$, then is $G$ abelian? This is an exercise I found in Jacobson's Basic Algebra. It is analogous to the question: If $G$ is a group that ...
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0answers
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A group with 3 Sylow 2-subgroup

Let $G$ be a finite group with $3$ Sylow $2$-subgroup(the number of Sylow $2$-subgroups $G$ are $3$), and let for every prime $p$ (not equal to $2$) Sylow $p$-subgroups are normal in $G$. I am looking ...
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Hall $\pi$-subgroups of normal subgroups

Let $A$ be a normal subgroup of $G$ such that $H\in$ Hall$_\pi$($A$) and $G/A$ is a $\pi$-group. Suppose that $H = H_1 \cap A$ where $H_1 \in$ Hall$_\pi$($G$). Show that $H^A = H^G$ where $H^G = ...
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1answer
102 views

Finding subgroups of the Real Numbers

Find a subgroup of $\left (\mathbb R -\{0\}, \times\right)$ with a finite number of elements, which is not just the trivial subgroup $\{1\}$. Find a subgroup of $\left(\mathbb R − \{0\}, ...
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Constructing an explicit isomorphism between automorphism group of bijective $F$-linear mappings and group of intertible $n \times n$ matrices

I'm going over some class notes: In the literature, sometimes a representation of $G$ over $F$ is defined as a pair $(V, \rho)$ where $V$ is a finite-dimensional $F$-vector space and $\rho: G \to ...
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Proving that something equals the commutator subgroup and conjugacy classes/normal subgroups

I've learned that the commutator subgroup is generated by the commutators. Now this says little about its elements (to me) because I don't see how they need to be commutators themselves. I'm ...
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Find $T_1(\langle (1,2,3,4,5,6,7,8,9) \rangle )$

$T_k(G)=\{g \in G : o(g) \, \,|\, \, p^k \}$ I am unsure what to do. Let that long permutation be $b$. Do we just find calculations of $b$ like $b^2$ or $b^{-1}$ or $b^b$ etc, that would give an ...
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29 views

Show $T_k(G)=G$

Suppose $G = C_n$ is an abelian $p$-group (so that $n = p^a$ for some $a$). Show that $T_k(G) = G$ if $a ≤ k$ and $T_k(G) = C_{p^k}$, otherwise. $T_k(G)=\{g \in G : o(g) \, \,|\, \, p^k \}$ If $a ...
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1answer
14 views

Prove $T_k(G)$ is a subgroup

$G$ is an abelian $p$-group. $T_k(G)=\{g \in G : o(g) \, \,|\, \, p^k \}$ for $k \ge 0$. Prove that this is a subgroup. For closed within multiplication, it is easy to see that $o(a)o(b)\,\, | \, ...
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Finding the order of an element

When we have permutation elements like $b=(12)(234)(1223)$ we can easily say that the order of each cycle is $2$, $3$ and $4$ respectively so the order of $b=\text{lcm}(2,3,4)$. When we have $C_n$, ...
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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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2answers
44 views

Finding subgroups of $D_8$

$$D_8=\{(),(1234),(13)(24),(1432),(13),(24),(14)(23),(12)(34) \}$$ Am I right to say that to find the subgroups, we have to make sure that the identity can be generated or is in the group and the ...
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1answer
43 views

Group with exactly two subgroups [duplicate]

Can someone give me an example of a group which has exactly $2$ proper, non-trivial subgroups? I appreciate that there is no context with this question but there's no really much else I can say.
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2answers
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Isomorphic normal subgroups [duplicate]

Let $N_1$ and $N_2$ be normal subgroups of G so that $N_1$ and $N_2$ are isomorphic. Is it true that then also $G/N_1$ is isomorphic to $G/N_2$?
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28 views

What is $(\mathbb{Z}\ast\mathbb{Z})\ast_\mathbb{Z}\cdots \ast_\mathbb{Z}(\mathbb{Z}\ast\mathbb{Z})$?

What is $(\mathbb{Z}\ast\mathbb{Z})\ast_\mathbb{Z}(\mathbb{Z}\ast\mathbb{Z})\ast_\mathbb{Z}\cdots \ast_\mathbb{Z}(\mathbb{Z}\ast\mathbb{Z})$ taken $n$ times? How can this be deduced easily? I think ...
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1answer
46 views

Presentation of $\pi_1$ of compact orientable surface by induction?

I need to prove by induction $\pi_1(\Sigma_g)= \left\langle a_1,b_1,\dots ,a_g,b_g\mid \prod_i [a_i,b_i] \right\rangle$. For genus 1 this holds since $\pi_1(T^2)\cong \mathbb Z\times \mathbb Z$. For ...