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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Give an example of a group G with a subgroup H and a prime p such that a Sylow p-subgroup of H is not a Sylow p-subgroup of G

I don't understand why this is possible. If H is a subgroup of G then you know the order of H divides the order of G and same with P a subgroup of H so how could p not be a Sylow p-subgroup of G?
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2answers
45 views

Is it possible to find two groups $G_1$ an $G_2$ of orders seven and eight and a morphism $f: G_1→G_2$ such that $|\operatorname{Im}f|=4$?

Is it possible to find two groups $G_1$ an $G_2$ of orders seven and eight and a morphism $f: G_1→G_2$ such that $|\operatorname{Im}f|=4$? How would you explain this?
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34 views

Assume $H\not\subset K\triangleleft G=⟨a⟩$ and a has finite order. prove $H\triangleleft G$

assume $H\not\subset K\triangleleft G=\langle a\rangle$ and $a$ has finite order. prove $H\triangleleft G$. I know that there is at most one subgroup of this group $G$ of every given order. But how ...
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2answers
24 views

if $P$ is a prime ideal of $O_K$, then $O_K/P$ is finite

let $P$ be a non-zero prime ideal of $O_K$, where $K$ is a number field(i.e. the degree $[K:\mathbb{Q}]$ is finite) then $O_K/P$ is finite. I'm working through a proof for this claim, however there is ...
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2answers
29 views

Let $p$ be a prime number; and $G$ a non abelian group or order $p^3$. Prove that $Z(G) = [G:G]$

I have already figured out that $|Z(G)| = p$, and that $G'=[G:G] \lhd G$. Also, $|G'| = p$ or $p²$ I suppose I'd have to prove that $G' \subset Z(G)$, but I've been trying and I have no idea how. ...
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2answers
39 views

Does this theorem have a formal name?

I am looking for a referable name for this theorem if one exists. The group $Z_n \times Z_m$ is isomorphic to $Z_{nm}$ if $n, m$ are relatively prime. Thank you.
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21 views

explicit zero 2-cocycle

Let $G$ be a group which acts linearly on a vector space of dimension $n$ over a field $k$. Denote by $\rho$ this representation and consider the associated adjoint representation $Ad\rho$ which is ...
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0answers
26 views

Order of any element divides the order of the group poof

Show that the order of any element $g \in G$ divides $n$, where $n$ is the order of a group $G$. So far I have shown that: $G$ - finite group and let $g \in G$, therefore, the order of the ...
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2answers
26 views

Extension of Naturals via Grothendieck Group Construction

So there is a way of extending the set $N$ of natural numbers with 0, equipped with ordinary multiplication, to its Grothendieck group, the group of integers with respect to addition. This group, ...
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1answer
31 views

Is it true that $H = H_1 \times H_2 \dots H_r$

Suppose that $G= G_1 \times \dots G_r$ be a decomposition of group $G$ into its normal subgroups. Let $H_i \leq G_i$ for every $i$.We know that for every $i \neq j$, we have $[G_i, G_j]=1$ and so ...
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2answers
22 views

There exists a isomorphism $f$ such that for every $k \in K \,$, $f(k) k^{-1} \in H$.

Let $G=H \times K$ and $G=H \times L$ be two decomposition of group $G$ into its normal subgroups. Prove that there exists a isomorphism $f: K \longrightarrow L$ such that for every $k \in K$, we have ...
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2answers
68 views

Proving $\mathbb{Q} \times \mathbb{Z_2} \ncong \mathbb{Q}$

How do I prove $\mathbb{Q} \times \mathbb{Z_2} \ncong \mathbb{Q}$? I know that they are not isomorphic because for each element in $\mathbb{Q}$, say $\frac{a}{b}$, there are two corresponding elements ...
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1answer
30 views

Normal subgroups and index problem

Let $G$ be a finite group and $H$ and $K$ subgroups such that $H \lhd G$ or $K \lhd G$. If $gcd(|K|;|G:H|)=1$, show that $K \subset H$. I think I could prove it for the case $H \lhd G$ Let $k \in ...
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1answer
53 views

Let $R$ be a commutative ring with $1 \ne 0$, and let $0 \ne e \in R$ be an idempotent element. Prove the following:

Let $R$ be a commutative ring with $1 \ne 0$, and let $0 \ne e \in R$ be an idempotent element. Note that $eR=\{er|r \in R\}$ is also a commutative ring with identity element $e$. (1) If I is an ...
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1answer
17 views

Why $c(a_1 a_2 … a_k)c^{-1}$ is the k-cycle $(c(a_1) c(a_2)… c(a_3))$?

If $a,b,c \in S_n$, why $c(a_1 a_2 ... a_k)c^{-1}$ is the k-cycle $(c(a_1) c(a_2)... c(a_3))$? (I need this to prove that two permutations are conjugate iff they have the same cyclic structure.)
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3answers
77 views

Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?

Let $G$ be a finite group, and $g$ be a an element of $G$. How could we go about proving $g^{|G|}=e$ without using cosets? I would admit Lagrange's theorem if a proof without talking about cosets can ...
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1answer
32 views

Proving a result concerning the size of orbits

I want to prove the following claim: Let $G$ be a finite group and $\alpha :G \to S_n$ a homomorphism. Then the size of every orbit of $\alpha(G)$ (considered as permutations on n letters) divides ...
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65 views

Prove this is a group

Prove $\mathbb Z_{235}$ is a group under multiplication. I know that a group must be closed under its operation, associative, have an identity element, and that every element must have an inverse. I ...
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2answers
29 views

Examples of Groups with only homomorphism sends every element of G to identity of H [on hold]

Give examples of non-trivial groups G and H so that the only homomorphism from G to H sends every element to the identity of H.
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1answer
24 views

Example of non-commuting conjugacy classes?

Let $x^G$ denote the conjugacy class of element $x$ in a group $G$ and $x^Gy^G = \{ab~:~a \in x^G, b\in y^G\}$ which contains, but may not equal, $(xy)^G$. Is there a simple example of a case where ...
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0answers
10 views

Relation between Jordan Normal Form and Irreducible Matrix Representations.

Ok so I have learned some very basic things about groups and matrix representations of groups. I have learned that it can be possible to find a "minimal basis" or "irreducible basis" for which a ...
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0answers
21 views

Which pairs of matrices generate $SL(2, \mathbb{Z})$?

I am trying to understand 2 element generating sets of $SL(2, \mathbb{Z})$ certainly these two: $$ \left(\begin{array}{cc} 1 & 1 \\ 0 & 1\end{array}\right) \text{ and } ...
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4answers
39 views

Showing that $\beta\alpha\beta^{-1}$ is a $5$-cycle

Let $G=S_n$, the symmetric group of degree $n$, where $n\geq 5$. Let $\alpha=(g~h~i~j~k)$ be a $5$-cycle in $G$. If $\beta$ is any element in $G$, show that $\beta\alpha\beta^{-1}$ is a $5$-cycle. ...
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3answers
27 views

Prove/disprove: Let $g\in G$ satisfy $o(g) = n$ and $g^m\in H$, where$ (n,m)=1.$ Then $g \in H.$

Here's a homework question: Prove or disprove: Let $g \in G$ satisfy $o(g)=n$ and $g^m \in H$ where $(n,m)=1$. Then $g\in H$. I've tested a couple of examples and couldn't find a counterexample. Any ...
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3answers
43 views

If G is a group with order 6, can all elements of g (except the identity) have order 2? If not, what is the underlying contradiction.

Trying to find a contradiction with out writing out a cayley table. So far, I understand that the group would be commutative. However I believe I fail to understand how closure is broken or in other ...
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1answer
22 views

Verifying that $\Gamma$ is a homomorphism and $\Gamma(G)\cong G/Z$.

Suppose $G$ is a group. Then the set of automorphisms of $G$, denoted by $\operatorname{Aut} G$, is a group under composition. Also, for $g\in G$, the map $\gamma_{g}\colon G\to G$ defined by ...
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3answers
29 views

Find an example of a group morphism

Find an example of a group morphism $f : G_1 \to G_2$ such that $H_1 ◅ G_1$ and $f(H_1)$ is not normal in $G_2$. How would you go about answering this question?
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2answers
44 views

If a group G is isomorphic to H, prove that Aut(G) is isomorphic to Aut(H)

If a group G is isomorphic to H, prove that Aut(G) is isomorphic to Aut(H) Properties of Isomorphisms acting on groups: Suppose that $\phi$ is an isomorphism from a group G onto a group H, then: ...
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1answer
15 views

No normal subgroup of a subgroup of $S_n$ imply the subgroup is the one of even permutations or consists of two elements

The following is an old exam question from a n introduction to group theory course: Let $G$ be a proper subgroup of $S_{n}$, $n\geq3$. Prove that if $G$ does not have any non-trivial normal ...
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1answer
27 views

free product with amalgamation is correspondingly a pushout

I'm trying to proof that the following diagram in the category of groups with $i_1$ and $i_2$ being inclusions is a pushout iff $G$ is the free product with amalgamation (up to isomorphism). It should ...
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1answer
19 views

If $G$ acts non-trivially on a set then it has a proper normal subgroup of finite index

The following is an old exam question from a n introduction to group theory course: Let $G$ be a group (possibly infinite) that acts non-trivially on a finite set $X$ - that is there are $g\in ...
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1answer
33 views

Prove the existence of order 4 subgroups of order 8 groups

I am participating in an Introductory course in groups and I have the following question: Let $G$ be a finite group of order $8$. Prove that $G$ has a subgroup of order $4$ and a subgroup of order ...
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0answers
34 views

How to find the subgroups of $\mathbb{Z}_{mn}^*$ isomorphic to $\mathbb{Z}_m^*$?

I already know that if m and n are coprime, then $\mathbb{Z}_{mn}^*$ is isomorphic to $\mathbb{Z}_m^* \times \mathbb{Z}_n^*$ using the Chinese remainder theorem. Now, Is it possible to describe ...
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0answers
37 views

Prove that these are not pairwise isomorphic

Prove $\mathbb Z_s$, $G_s$ (the group of symmetries of the square) and the quaternion group $Q$ are not pairwise isomorphic. How would you go about proving. Seems quite difficult. I know that none of ...
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1answer
29 views

Split extension for semi-direct and direct products. Can a split extension be exact?

Question: Can there be a split extension with $G=N\times H$? Also can a split extension be exact? If so, when?(From bottom for increased context) Definition of exact sequence (in the context of ...
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2answers
49 views

Finite subgroup of $\mathbb C^{\times}$

I was trying to show that every finite subgroup of $\mathbb C^{\times}$ is equal to $G_n$ (the nth roots of unity) for some $n \in \mathbb N$ without invoking Lagrange's theorem, I got stuck at one ...
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1answer
36 views

sylow p-subgroups of S3, S4 and S5

I want to find the sylow p-subgroups of S3, S4 and S5, but I don´t understand the iterated wreath products, can I help me?
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Let $R$ be a Euclidean domain with degree function $\varphi$ and $R$ not a field. If $a \ne 0$ and $b \ne 0$ be two elements in $R$…

Let $R$ be a Euclidean domain with degree function $\varphi$ and $R$ is not a field. Prove the following: (1) Let $a \ne 0$ and $b \ne 0$ be two elements in $R$. Suppose that $a\mid b$ and $b \nmid ...
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1answer
20 views

Show that a cycle of length $p$ and a cycle of length $q$ in $S_n$ are conjugate if and only if $p = q$.

Show that a cycle of length $p$ and a cycle of length $q$ in $S_n$ are conjugate if and only if $p=q$. First of all, I'm a bit confused about the meaning of '... are conjugate'. Does this mean that ...
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0answers
18 views

Implications of the order of subgroups, given the order of the parent is a composite number

If you are told that the order of a group $G=x$ such that $x=x_1x_2...x_n$ is the prime factorisation of $x$. Does this imply that subgroups of order $x_i ; i \in \{1, 2,...,n\}$ must all exist?
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26 views

Let $ H $ be a subgroup of $ G $ auch that $ | G:H | $ is a $ \pi $-number [on hold]

Let $ H $ be a subgroup of $ G $ such that $ | G:H | $ is a $ \pi $-number. If there is a nilpotent subgroup $ K $ of $ G $ such that $ G = HK $. let $ K = K_{\pi^{\prime}}K_{\pi} $, where $ ...
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1answer
48 views

How did this happen? [on hold]

If $S$ be a finite group of order $n$ , then the set $A(S)$of all one to one mapping from S into itself has cardinality $n!$. How $ A(S)=n!$ ?
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Proving $Ind_H^G1_H=\pi_X$

Lemma Let $\psi=1_H$ the principal character of $H$, then $Ind_H^G1_H=\pi_X$, the permutation character of $G$ on the set $X$ of left cosets of $H$ in $G$. Proof Let $\{t_i \}$ form a tranversal. ...
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0answers
34 views

Composition of polynomials over finite fields

Consider the set of polynomials of degree at most $n$ over a finite field $k_q$ with $q$ elements where $q$ is prime: $$ P_{n,q} = \left\{ x + c_2 x^2 + \cdots + c_n x^n:\ c_i \in k_q \right\}. $$ It ...
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1answer
33 views

Prove that if G has a faithful complex irreducible representation

I am struggling with the proof that if G has a faithful complex irreducible representation then $Z(G)$ is cyclic: Let $\rho:G \rightarrow GL(V)$ be a faithful complex irreducible representation. ...
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1answer
43 views

Finite matrix groups as subgroups of $S_n$.

I have heard that all finite subgroups are isomorphic to a subgroup of $S_n$. I was thinking about examples of this. In particular I would like to know how this works for certain matrix groups. The ...
2
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3answers
57 views

Algorithm to generate an arbitrary matrix of special linear group $SL(2,\mathbb{Z})$

I have a given $2\times 2$ special linear matrix, for example \begin{equation} m=\begin{pmatrix} 55 & 8469 \\ 1 & 154 \end{pmatrix} \end{equation} and I would like to get the generating form ...
0
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1answer
30 views

Is the intersection between a subgroup, and a normal subgroup, normal in the parent group?

Given $H, N \subset G$ and $N \lhd G$ is there some underlying fact or theorem for why $H \cap N$ would or would not be normal in $G$? My reasoning would be that it would be normal to $G$ as; $\forall ...
5
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3answers
65 views

About a relation of non-discernability between (classes of) finitely generated groups.

Let $G$ be the set of finitely generated groups up to isomorphism. Now define $B$ and $C$ two finitely generated groups to be not discernable if one can find a finitely generated group $A$ such ...
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1answer
17 views

Finding the two different conjugacy classes in $A_n$ after splitting criterion

Suppose we have a group $A_n$ for some $n$ (maybe take $A_5$ as an example). We find the conjugacy classes of $S_n$ which are determined by cycle type. Then we use the splitting criterion ...