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

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A question about the dihedral group $D_n$

Let the dihedral group $D_n$ be given by elements $a$ of order $n$ and $b$ of order $2$, where $ba = a^{-1}b.$ (a) Show that $a^{-m}= a^{n-m}$ for all integers $m.$ Definition: Let $n$ greater than ...
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2answers
68 views

Cosets of Group theory

Consider the group $(\mathbb{Q}, +)/(\mathbb{Z}, +)$, the group of rationals (under addition) modulo the subgroup of integers. So an element of this group is a coset $a + \mathbb{Z}$ where a is a ...
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1answer
90 views

Module, vector space and abelian group

please can someone help me to prove this two properties $\bullet$ If K is a field, then the concept of K-vector space (a vector space over K) and K-module are identical. $\bullet$ The concept of a ...
6
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3answers
239 views

The group of invertible elements of $\mathbb F_{p}[x]/(x^m)$ is not a cyclic group.

I am stuck in a question about finite fields and would like to ask you for some help. Given an integer $m\geq 2$ and $p$ a prime number, show that $(\mathbb F_{p}[x]/(x^m))^{\times}$ (the group ...
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1answer
190 views

Infinite Dihedral Group.

Let $D_{\infty}= \langle x,y \mid x^2=y^2=1\rangle$ be the infinite dihedral group. Are the following statements true? Since $G$ is not torsionfree, $\mathbb{Q}[G]$ is not a domain. $D_{\infty}$ is ...
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2answers
63 views

how to find the elements of additive Group - $\mathbb{Z_7^+}$

I am given this additive Group G=$\mathbb{Z_7^+}$ I tried to find all its elements and I did: $$gcd(1,7) = 1 \\ gcd(2,7) = 1 \\ gcd(3,7) = 1 \\ gcd(4,7) = 1 \\ gcd(5,7) = 1 \\ gcd(6,7) = 1 \\ ...
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1answer
64 views

graphs and groups

In many papers we see a group which is constructed a graph from it and captures some information about the group. There are several ways of doing this, non-commuting graph, power graph, cayley ...
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1answer
41 views

Is $\text{fix}_\Omega(G_\alpha)$ a block of imprimitivity when $G$ is infinite?

Let $G$ be an infinite transitive permutation group acting on a set $\Omega$. Is $\text{fix}_\Omega(G_\alpha)$ a block of imprimitivity for $G$ in $\Omega$? $G_\alpha$ is the set of elements of ...
3
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0answers
76 views

Find the fundamental group and the Alexander polynomial

I would like to find the Alexander polynomial of the link $L$, described below. Let $K(q,r)$ be the $(q,r)$-torus knot embedded on a torus $V$. Inside the torus $V$, consider a smaller solid torus ...
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1answer
71 views

Does every non-trivial subgroup of $S_9$ containing an odd permutation necessarily contain a transposition?

Does every non-trivial subgroup of $S_9$ containing an odd permutation necessarily contain a transposition? Here $S_9$ denotes the group of all permutations (i.e. bijections with itself) of the ...
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1answer
200 views

Normal Subgroups of a Nilpotent Group

Let G be a finite nilpotent group with order n. Is it necessarily true that for all divisors m of n, G contains a normal subgroup H such that ord(H)=m? Why or why not? I was able to show that G always ...
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7answers
349 views

What is a coset?

Honestly. I just don't understand the word and keep running into highly technical explanations (wikipedia, I'm looking at you!). If somebody would be so awesome as to explain the concept assuming ...
2
votes
1answer
62 views

Two subgroups of a group

Let $A$ and $B$ be two subgroups of a group $G$. If $|A| = p$, a prime integer, then show that either $A ⋂ B = \{e\}$ or $A$ is a subset of $B$.
2
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1answer
49 views

Need Help in subgroups

Let $H$ and $K$ be subgroups of a finite group $G$ such that $|H|>\sqrt{|G|}$ and $|K| > \sqrt{|G|}$. Show that $|H \cap K| >1$
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3answers
171 views

Left coset of some subgroup is right coset of other subgroup

Let $G$ be a group. If a subset $A$ is a left coset of some subgroup of $G$, then show that $A$ is a right coset of some subgroup of $G$.
4
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1answer
75 views

Necessary and Sufficient conditions to generate $S_n$

I have a homework question that asks "Find necessary and sufficient conditions on $1 \leq i < j \leq n$ so that $(i \, j)$ and $(1 \, 2 \, \dotsc \, n)$ generate $S_n$." Here is what I have done ...
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0answers
206 views

Values attained by $|G/Z(G)|$?

So I was working through some problems in a book on $p$-groups and noticed that $p$-groups have some really nice properties. So I started computing what the values of $|G/Z(G)|$ for $p$-groups. I ...
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4answers
151 views

Can someone explain automorphisms to me?

So I know the definition of an automorphism is an isomorphism that maps from a group to itself. How can an element of an automorphism map to something besides itself?
2
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2answers
89 views

Residually finite nilpotent group

It is known that every finitely generated nilpotent group is residually finite. Why finitely generated hypothesis is essential?
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1answer
50 views

Cosets and order

Let $H$ and $K$ be a subgroups of a group $G$, of orders |H|=2 and |K|=5. Some cosets of H may contain elements of K. How many such cosets could there be ?
4
votes
1answer
77 views

On group theory terminology

Let $G$ be a finite group. Consider the next number $$m(G):=\min\{m\in\mathbb{N}\mid G\ \text{can be embedded into}\ S_{m}\}.$$ It is obvious that Cayley's theorem yields $m(G)\leq |G|$. My ...
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0answers
79 views

Existence of a particular element in $U(n)$ the unit group of integers.

$U(n)$ is the group of all the units in $\mathbb Z_n$. If $n>2$, prove that there is an element in $U(n)$ such that $k^2 = 1$ and $k\neq 1$. (From the 2013 edition of Abstract Algebra: Theory ...
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1answer
105 views
3
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1answer
76 views

About triplets of integers inducing commutativity in groups

It is well-known that if $i\in\mathbb Z$, then any group $G$ is abelian provided that $(ab)^k=a^kb^k$ holds for $k=i,i+1,i+2$ and for all $a,b\in G$ (see for example this question). Are there other ...
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1answer
33 views

Relation between a set being closed under a binary operation and the set being a group under that binary operation

If a set $S$ is not closed under some binary operation $\star$, is it true that $S$ cannot be a group under $\star$?
4
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1answer
48 views

If $G$ is a group, and $a, b \in G$, show that if $a^{-1} b^{2} a = b^{3}$ and $a^{2} = 1$, then $b^{5} = 1$

I've been playing with this one for a while and I can't seem to get any closer to the solution. Does anyone have any suggestions or a hint?
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1answer
99 views

find all the units of $R = \mathbb{Z}[\sqrt{-n}] = \{ a + b\sqrt{-n} \mid a,b \in \mathbb Z\}$

Let $n$ be a natural number. Define the ring $R = \mathbb{Z}[\sqrt{-n}] = \{ a + b\sqrt{-n} \mid a,b \in \mathbb Z\}$. Find all the units in $R$. There is a hint that we can define $H: R \to ...
3
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1answer
70 views

Show that $\mathbb{Q}(\zeta)$ is the Splitting Field for $x^n - 1 \in R_n[x]$

Let $R_n = \{\bar{x}$ modulo $n : (x,n) = 1\}$ which forms a group under multiplication. Let $p(x) = x^n - 1 \in \mathbb{Q}_n[x]$ have roots $\zeta_1, \zeta_2, \ldots , \zeta_n$. Prove that there is ...
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2answers
180 views

Infinite group must have infinite subgroups.

Prove that an Infinite group must have subgroup with infinite elements. I know that if group was cyclic order of the generator is infinite and there are infinite number of divisors.
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0answers
190 views

Show $G$ is isomorphic to the multiplicative group $\mathbb{R}^\times$

Let $G=\{x\in\mathbb{R}\mid x>0 \text{ and }x\neq 1 \}$ and define $*$ on $G$ by $a*b=a^{\ln b}$. Show $G$ is isomorphic to the multiplicative group $\mathbb{R}^{\times}$. I need to ...
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300 views

Cyclic group of prime order [closed]

If G be a cyclic group of prime order p, prove that every non identity element of G is a generator of the group.
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1answer
122 views

Intersection of distinct, primed-ordered subgroups is trivial. [closed]

$H$ and $K$ are different subgroups of a group $G$ such that $o(H) = o(K) = p$ where $p$ is prime. Show that $H \cap K = \{e\}$. Deduce that if $G$ has exactly $m$ distinct subgroup of prime order $p$ ...
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1answer
202 views

Finding the number of elements of order $2$ in the given group

How many elements of order $2$ are there in the group of order $16$ generated by $a$ and $b$ such that $o(a)=8$ and $o(b)=2$ and $bab^{-1}=a^{-1}$ I basic thing i do not understand is that order of ...
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0answers
54 views

Upper central series in Coclass Theory.

It is proved by Aner Shalev, that for any finite $p$-group of coclass $r$(and sufficiently large order), there is some severe restrictions on lower central series $(\gamma_i(G))$. For instance, ...
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0answers
54 views

On some endomorphisms of finite groups of odd order.

Let $G$ be a group of odd order. It is known that if every central automorphism of $G$ acts trivially on the center, then $G$ is purely non-abelian, this amounts to saying that every central ...
6
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1answer
116 views

Semigroups defined by the subsets of groups

In a group $G$, the non empty subsets form a semigroup (with identity) under the usual multiplication $ST=\{st \mid s\in S, t \in T\}$. This semigroup seems to be very rich of information, for ...
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2answers
178 views

Prove that an automorphism group is a normal subgroup

Let $G$ be a group, let $T$ be an automorphism of $G$, and let $N$ be a normal subgroup of $G$. Prove that $T(N)=\{T(x) \mid x\in N\}$ is a normal subgroup of $G$. I would prefer a hint to get ...
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4answers
600 views

Need to prove that (S,*) defined by the binary operation a*b = a+b+ab is an abelian group on S = R \ {1}

So basically this proof centers around proving that (S,*) is a group, as it's quite easy to see that it's abelian as both addition and multiplication are commutative. My issue is finding an identity ...
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0answers
78 views

Structure of finite abelian group

I am trying to solve this problem, and I think I should use the structure theorem for finite abelian groups, but i can't really figure this out. Let $G$ be a finite abelian group. Prove there is a ...
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2answers
100 views

Abstract Algebra Group

If a and b are distinct group elements, prove that either $a^{2}\neq b^{2}$ or $a^3\neq b^3.$ I tried doing the operation by inverses to get the identity, but that does nothing.
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1answer
67 views

Alternating Group

Suppose that $H$ is a subgroup of $S_n$ of odd order. Prove that H is a subgroup of $A_n$. How can I solve this problem without using Cayley's Theorem? So far, I understand that H contains both even ...
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3answers
44 views

What are the various ways to form new subgroups out of existing ones?

Let $G$ be an abelian group. Let $A,B,C,\leqslant G, \ \ (A_i)_{i\geq 1} \subset G$ be subgroups. Then from what I know you can do the following: $A\cap B$ If $A_i \subset A_{i+1} \forall i$, ...
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0answers
35 views

Graph theory: Linking graph characteristics and minimal cut

I'm currently working on a research involving Graph theory. More specifically, I would like to make an analytical or theoretic connection between different characteristics of the graph (e.g. size, ...
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1answer
138 views

Proving that a metric space is a group

I'm stuck on this relatively hard problem. Let $G$ be a non-empty set, $d$ a distance on $G$ and $\cdot$ an associative operation on $G$ $\cdot$ is such that $$\forall a \in G , \forall x \in G ...
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1answer
88 views

Group Theoretical Classes

$\textbf{Definitions}$ Let $\mathfrak X$ be a class of groups (such as the one of cyclic groups). Let $\textbf H\mathfrak X$ be the class of factor groups of $\mathfrak X$-groups. Let $\textbf ...
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1answer
110 views

Given $G$ is a group and $a,b\in G $ and $ab=ba$. Prove…

$ab^n= b^na\;\; \forall n \in \mathbb{Z}$ I have been able to prove this for $n=0$ and for a positive integer (using induction). But for $n$a negative integer, I'm not able to prove it: $n=-m$ for ...
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1answer
38 views

Question about definition of generator for dihedral groups?

Dummit and Foote mention that a relation for the dihedral group is $rs = sr^{-1}$. Now, I have interpreted the statement to mean $r$ is a rotation of $\frac{2\pi}{n}$ radians and $s$ is an ...
0
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1answer
24 views

Modules over a group presented via a free group.

Say $G$ is presented via a free group $F$ freely generated by $S=\{s_i, 1=1,2,\dots\}$. Then $\pi:F \rightarrow G$ the canonical projection. Let $R$ be any commutative ring. Can we follow that any ...
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2answers
152 views

Elementary manipulation with elements of group

Let $(G,*)$ be a group with identity $e$ , let $a,b∈G$ such that $a*b^3*a^{-1}=b^2$ and $b^{-1}*a^2*b=a^3$ , then how do we prove that $a=b=e$ ?
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Is there a non-trivial binary operation on the set of subgroups of a finite group that distributes with intersection?

Let $G$ be a finite abelian group. Then is there a $\cup$-like operation we will call group union such that it distributes over subgroup intersection? Let $\mathcal{H}(G)$ be the set of subgroups of ...