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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Show that order of $a^k$

I have seen this everywhere but how do I show that : if $a$ is an element of order $n$ in a group $G$ then the order of $a^k$ is $\frac {n}{(n,k)}$. We know, since $a$ has order $n$, then $a^n = 1$
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1answer
45 views

A line in a proof regarding nth power residues

I would appreciate help understanding this highlighted line in a proof in Ireland & Rosen (p. 45). I don't know much group theory although I know the residue classes $\pmod m$ form a ...
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1answer
16 views

Subsets of $G$-sets with sharply transitive $G$-action

Let $G$ be an infinite group acting sharply transitively on a set $X$. Let $Y\subset X$ be a proper subset. Is there a subgroup $H\leq G$ which acts sharply transitively on $Y$ ? I think this is ...
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2answers
36 views

Let $G$ be a group of order 1210 with a subgroup $H$ of order 121. Show that every element of order 11 is in $H$

Let $G$ be a group of order 1210 with a subgroup $H$ of order 121. Show that if $a\in G$ has order 11, then $a\in H$ This was a question on a test I just took and even though I spent almost all ...
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4answers
109 views

What is $\mathbb{R}/\mathbb{Z}$ going to be?

Still kind of kind of not getting the hang of this. When we see $\mathbb{Z}/\mathbb{2Z}$ for instance, I can relate it to the set of integers modulo $2$. I look at $\mathbb{R}/\mathbb{Z}$ and I am ...
2
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2answers
53 views

Product group isomorphism

Let $H$ and $K$ be subgroups of a group $G$, and let $f:H\times K \to G$ be the multiplication map, defined by $f(h,k)=hk$. Show that $f$ is an isomorphism from the product group $H \times K$ to $G$ ...
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3answers
30 views

Two normal subgroups and isomorphism theorem

Question Let $N_1$ and $N_2$ be normal subgroups of $G$. Prove that $N_1N_2/(N_1\cap N_2) \cong (N_1N_2/N_1)\oplus (N_1N_2/N_2)$. I think the homomorphism must be $\phi : N_1N_2 \to ...
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3answers
43 views

Group of order $p^2$ [duplicate]

Let $G$ be a group of order $p^2$ where $p$ is a prime. I want to show that either $G$ is cyclic or is the product of two cyclic groups of order $p$. After some work, the problem reduces to showing ...
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1answer
61 views

Any nonabelian group of order 12 is isomorphic to A4, D6, or Z3 x Z4 [closed]

Can someone show me the proof for : Any nonabelian group of order 12 is isomorphic to $D_6$, $A_4$, or $\mathbb{Z}_3 \times \mathbb{Z}_4$ Ive seen a few proofs where this is included in also ...
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1answer
35 views

How to proof that $ba=a^6b$ in the dihedral group $D_{14}$ [on hold]

How do you prove that $ba=a^6b$ in the dihedral group $D_{14}$ ?
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1answer
23 views

What is the binary operations in $\mathbb{D}(n)$

I am using Joseph A gGallian for group theory.I came across something named Dihedral group $\mathbb{D}(n)$. The author uses only Caley table and does not describe the binary operation of this group ...
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0answers
25 views

How does the Whitehead quadratic functor act on morphisms?

I am trying to understand the universal quadratic functor $\Gamma$ that appears on the Whitehead exact sequence (J. Whitehead, A certain exact sequence, Annals of Mathematics 52 (1950)) particularly ...
2
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1answer
319 views

Group Order and Least Common Multiple

Let $G_1,G_2,...,G_n$ be groups. Show that the order of an element $(a_1,a_2,...,a_n)$ $\in$ $G_1 \times G_2 \times\cdots\times G_n$ is lcm($o(a_1),...,o(a_n))$. I know I need to use the fact ...
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1answer
19 views

Von Dyck groups that are conjugated.

Let us consider the Von Dyck groups $$ D(a,b,c)=\langle x,y,z\mid x^{a}=y^{b}=z^{c}=xyz=1\rangle $$ and $$ D(a'.b',c')=\langle x,y,z\mid x^{a'}=y^{b'}=z^{c'}=xyz=1\rangle. $$ Suppose $$ ...
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1answer
265 views

Prove that $G$ has a element whose order is least common multiple of $m$ and $n$. [duplicate]

Let $G$ be an abelian group and suppose that $G$ has elements of order $m$ and $n$ respectively. Prove that $G$ has an element of order $\mathrm{lcm}[m,n]$
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3answers
768 views

Order of element equal to least common multiple [duplicate]

Let $G$ be a group, and $a,b\in G$. Suppose $\operatorname{ord}(a)=m, \operatorname{ord}(b)=n$, and that $ab=ba$. Prove that there is an element $c\in G$ such that ...
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1answer
19 views

Showing a translation group is a normal subgroup of an affine group

Let V be an n-dim vector space over the field $\mathbb{F}.$ $A \in GL\left ( n,\mathbb{F} \right )$ and $v \in V$ Define the affine transformation $t_{A,v}$: $V\rightarrow V$ $x \mapsto xA+v$ ...
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0answers
41 views

How many Isomorphisms are there from $\mathbb{Z}_3\oplus\mathbb{Z}_5$ to $\mathbb{Z}_{15}$

How many Isomorphisms are there from $\mathbb{Z}_3\oplus\mathbb{Z}_5$ to $\mathbb{Z}_{15}$ We have generator of $\mathbb{Z}_3\oplus\mathbb{Z}_{15}$ is $1$ and there $\phi(15)=8$ generators of ...
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1answer
27 views

Prove that if $H \cong \mathbb Z$ or $H \cong \mathbb Z_n$ then $H = ⟨g⟩$ for some $g ∈ G$.

I'm studying for my final and I came across this homework problem that I had previously done but I don't remember how to do it anymore. It is as follows ($G, H$ are groups): Suppose that $H \le G$. ...
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38 views

Proof of the fixed point theorem in group action.

$\textbf{Fixed point theorem:}$ Let $G$ be a $p$-group and let $S$ be a finite set on which $G$ operates. If the order of $S$ is not divisible by $p$, there is a fixed point for the operation of $G$ ...
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1answer
12 views

Does the set of all 3x3 echelon form matrices with elements in R form a subspace of M3x3(R)? Same question for reduced echelon form matrices.

Screenshot of the past exam question Firstly, the zero 3x3 matrix denoted as A is both in echelon and reduced echelon form since it satisfies both definitions respectively. $$ A = ...
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1answer
30 views

Generators of two groups with prime order $p$ already induce all the generators of the product group $G \times H$

Let $G = \langle g \rangle, H = \langle h \rangle$ be two cyclic groups (with $g \in G, h \in H$), both of them of order $p \in \mathbb{N}$, where $p$ is a prime number. I now want to show that ...
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1answer
20 views

meaning of a subgroup normalizes . [duplicate]

I know that if a subgroup A is normal subgroup of the subgroup B,then B is contained in the normalizer of A . But I don not know what is the meaning of A normalizes B ?
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1answer
19 views

Discrete action on Lie groups

Given a Lie group $G$ and a discrete subgroup $\Gamma$ of $G$. Why is the action of $\Gamma$ on $G$ properly discontinuously?
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32 views

Proof check: a group $G$ with presentation $[a,b\mid ab=e]$ is isomorphic to $\mathbb{Z}$

Given that $ab=e$, we know $b =a^{-1}$. Since $G = \langle a,b\rangle$ this implies $\langle a,b\rangle=\langle a,a^{-1}\rangle=\langle a\rangle=G$. The presentation rewritten in terms of $a$ is ...
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1answer
46 views

Classify groups of order 100 [on hold]

So I am currently trying to Classify all groups of order 100 through an extensive proof; and this is as far as I have gotten so far, wondering how to go beyond the fact that both squares (Z4 & ...
0
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1answer
22 views

Why is $\bar{B^t} \bar{A^t} = \overline{(AB)^t}$ true? [on hold]

In class I've encountered the following thing: $\bar{B^t} \bar{A^t} = \overline{(AB)^t}$. I don't understand why that's true.
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2answers
49 views

Prove that $(H,\circ)$ is a subgroup of the group $(G, \circ)$

Question: Let $(G, \circ)$ be a group and $H$ be a non-empty subset of $G$. A relation $\rho$ defined on $G$ by $$a\,\rho\ b\quad \text{if and only if}\quad a\circ b^{-1}\in H$$ for $a,b\in G$, is an ...
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1answer
40 views

$H$ is a subgroup of $G$ with finite index. Prove that G has finitely many subgroups of form $xHx^{-1}$

$H$ is a subgroup of $G$ with finite index. Prove tat $G$ has finitely many subgroups of form $xHx^{-1}$. Let $h\in H$, $x\in G$ Since H is a subgroup of G $h \in G$ $\rightarrow he \in G$ ...
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1answer
16 views

Factoring a homomorphism through the quotient by a normal subgroup contained in the kernel

Suppose $f\colon G\to G^{\prime}$ is a group homomorphism. Let us denote the groups additively. It is well know that such a homomorphism always can be 'factored through' the quotient ...
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1answer
45 views

Fulton and Harris exercise 5.7

I would like to know if I'm on the right track: We're trying to derive the character table for $PGL(2,q=p^k)$ from $GL(2,q)$. The table for $GL$ is given in the text. It gives the hint that the ...
3
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2answers
41 views

The triangle group $(\alpha,\alpha,\alpha)$ is a subgroup of the triangle group $(3,3,\alpha)$

This answer made me wonder if there is a geometrical way to prove that the triangle group $(\alpha,\alpha,\alpha)$ is a subgroup of the triangle group $(3,3,\alpha)$. In other words, how can we ...
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0answers
19 views

Showing the affine transformation is well-defined

Let V be an n-dim vector space over the field $\mathbb{F}.$ $A \in GL\left ( n,\mathbb{F} \right )$ and $v \in V$ Define the affine transformation $t_{A,v}$: $V\rightarrow V$ $x \mapsto xA+v$ ...
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2answers
23 views

Product of a normal subgroup with a subgroup generated by two subgroups

Let $G$ be a group. Suppose that $N\unlhd G$ and $A,B \leq G$. I want to show that $\langle A, B\rangle N = \langle AN, BN\rangle$. Clearly, $AN \leq \langle A, B\rangle N$ and $BN \leq \langle A, ...
1
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1answer
74 views

For a group $G$ and subgroup $H$, is $a \sim b \iff a^{-1}b\in H$ an equivalence relation even when $H$ is not normal?

Is it true or false that defining a relation on the group $G$ based on the definition $a \sim b$ if and only if $a^{-1}b\in H$ defines an equivalence relation regardless of whether $H$ is a normal ...
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1answer
22 views

How does one find the automorphism group of the following groups?

An automorphism of a group G is an isomorphism of G in itself. I am trying to find the automorphism groups of: $\mathbb{Z}; \mathbb{Z}/p\mathbb{Z}$ p prime; $\mathbb{Z}/6\mathbb{Z}$ I know that any ...
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0answers
18 views

Proof That For a Group $G$, All Automorphism of G Is Defined by The Image of The Generating Set Of G.

Let (G,*) be a group. Let $\Phi: G \rightarrow G$ be the automorphism of $G$. I want to show that all automorphism of $G$ is defined by the image of de generating set of $G$. My proof Let ...
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0answers
25 views

PGL(2,c) is isomorphic to PSL(2,c)

Definition: Projective special linear group $PSL\left ( n,\mathbb{F} \right )=\frac{SL\left ( n,\mathbb{F} \right )} {\left ( Z\left ( GL\left ( n,\mathbb{F} \right ) \right )\cap SL\left ( ...
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0answers
17 views

Language used in projective linear group

In lectures and text on topic of projective linear group, I hear and see the word "factor out" or "quotient out" thrown around a lot. What is the word supposed to mean? If this is vague, I can ...
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1answer
22 views

Iff $\{s t^{-1}: s, t \in T \} = G$ for a group $G$ and a nonempty subset $T$, then each right coset $Ng$ of $G$ is already trivial

Let $G$ be a group and $T \subseteq G, T ≠ \emptyset$. We now want to consider the set $T T^{-1} := \{s t^{-1}: s, t \in T \}$. I now want to show that $\langle T T^{-1} \rangle ≠ G $ iff there ...
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0answers
7 views

Norm estimates on Markov operator.

Let $G$ be a group, and let $S$ be it's finite symmetric generating set. Assume that Markov operator is defined as $$M(f)=\sum\limits_{s \in S} g.f $$ Obviously, $f$ can be a function in $l_p$. ...
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3answers
48 views

Group operation is well defined

One of the most common manipulations performed when working with group equations is left or right multiplication, i.e. if you have a group $G$ with $a,b,c \in G$ and you have something of the form $a ...
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2answers
55 views

Abstract Algebra, group theory

It is given that order of exactly eight elements of group G is 3. We have to find the number of subgroups of order 3? I just know it has to do something with cyclic groups And I'm guessing the ...
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2answers
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Can RUBIK's cube be solved using group theory?

Can RUBIK's cube be solved using group theory? If yes, how can we use it to solve a $2\times2$ Rubiks Cube?
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2answers
27 views

How do I calculate the group of automorphisms of a cyclic group [on hold]

Definition An automorphism of a group G is an isomorphism of G in itself: How do I calculate the group of automorphism of the following groups: $\mathbb{Z}$ $\mathbb{Z}/p\mathbb{Z}$ $p$ prime ...
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0answers
15 views

If $P/Z(P)$ has nilpotency class $k$ then $P$ has nil potency class $k+1$

I would like to prove the following fact. Let be $P$ a p-group, with $|P|=p^n$ , $p$ a prime number. I suppose that $P/Z(P)$ is a nilpotent group with class of nilpotency $k$ and I want to prove ...
0
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1answer
26 views

Element of a subgroup generated by two subgroups

Let $G$ be a group and suppose that $A$ and $B$ are subgroups of $G$. I want to know what a typical element in the subgroup $\langle A, B\rangle$ looks like?
1
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1answer
29 views

Quotient group with normal subgroup dividing the order of another group [duplicate]

Let G be a group with subgroup H and let $\Omega$ be the set of right cosets of H in G. Show that if G is a group with a subgroup of index n then G has a normal subgroup with index dividing n! ...
2
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0answers
35 views

Irreducible components of tensor product representations.

Let $(\rho,V)$ be an irreducible representation of a finite group $G$, and let $W$ be a vector space. Then clearly $(\rho\otimes\text{Id}_{W},V\otimes W)$ is also a representation of $G$. I would like ...
-2
votes
1answer
27 views

On finite abelian p-groups all of whose maximal subgrops are cyclic [on hold]

Let $G$ be a finite non-cyclic abelian $p$-group such that all of whose maximal subgroups are cyclic. Then what can we say about the structure of $G$?