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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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
81 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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40 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 $\star$?
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1answer
50 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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147 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
86 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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326 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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251 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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671 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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206 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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2answers
358 views

Finding the number of elements of order $2$ in a 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 $bab^{-1}=a^{-1}$? The basic thing i do not understand is that ...
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59 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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62 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
135 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 ...
4
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2answers
286 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
2k 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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100 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 ...
2
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2answers
181 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
78 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
45 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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39 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
178 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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92 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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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
55 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 ...
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1answer
27 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
156 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 ...
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228 views

Proving $G$ is a group if it is closed under product and both cancellation laws hold.

I would be grateful if someone pointed out an error in my answer. Prove that a if finite set $G$ is closed under an associative product and that both cancellation laws hold in $G$, then it is a ...
2
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2answers
112 views

When is mapping $g$ to $g^{-1}$ a group homomorphism?

When is mapping $g$ to $g^{-1}$ a group homomorphism? It just means that the map maps the identity to the identity and inverses to inverses. So does that mean it's only a homomorphism if the inverse ...
2
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2answers
66 views

$K$ and $gKg^{-1}$ have the same cardinality

How do I show that $K$ and $gKg^{-1}$ have the same cardinality for some $K$ that is a subset of $G$? I thought I might use Lagrange's theorem, but it's obvious that cosets have the same cardinality, ...
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Is this a theorem?

There exists a group homomorphism of a group $G$ into a group $H$ if and only if there exists a direct sum $G \approx \bigoplus_{i\in I} H_i$ that involves $H$, i.e. $H_i = H$ for some $i$. Proof: ...
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56 views

Find a homomorphism from $\mathbb{Z}$ to $SO(2,\mathbb{R})$

If we pick some element $x$ in the $SO(2,\mathbb{R})$, how do we write out the homomorphism explicitly?
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1answer
109 views

Show that there exists two Sylow $p$-subgroups $P$ and $Q$ such that $[P:P\cap Q] = [Q:P\cap Q] = p$

Let G be a group with the number of Sylow $p$-subgroups different from 1 mod $p^2$. Show that there exists two Sylow $p$-subgroups $P$ and $Q$ such that $[P:P\cap Q] = [Q:P\cap Q] = p$. First, ...
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217 views

How can we determine associativity of a binary structure from its Cayley table?

Suppose $S$ is a finite set with a binary operation $*$ given by a Cayley table. While the commutativity of $*$ can be determined on the basis of the symmetry of the table across the upper-left to ...
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Alternate Proofs to this Question?

The question asks: "Does there exists a non-abelian group of order 2012?" My answer is yes, an example of which is the dihedral group $D_{1006}$. I'm curious, though, if anyone can give me a ...
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1answer
49 views

Example where $\{g \in G \mid \phi(g) = g^{-1}\}$ is not a subgroup for $\phi \in {\rm Aut}(G)$

Let be $G$ a group and $\phi \in \operatorname{Aut}(G)$. Give an example in which the subset $$I(G,\phi)=\{g\in G \mid \phi(g)=g^{-1} \}$$ is not a subgroup of $G$. Is easy to see ...
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P p-sylow and H a p-subgroup of G

Proposition. If P is a p-sylow of G and H a p-subgroup then $H ∩ N_G(P ) = H ∩ P $ First I try to prove that If $T=H ∩ N_G(P )$ then $TP$ is a p-subgroup of G, how can i prove this and conclude ...
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2answers
114 views

Does the language of group theory need a constant?

I would like to verify the understanding of my claim. In Model Theory: An Introduction by David Marker Example 1.2.5 defines the language of the theory of groups as follows to be $\mathcal{L}=\{., ...
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108 views

Reference request for Lorentz group and unitary representations

More precisely, I often read or listen that Lorentz group has not (non trivial) unitary finite dimensional irreducible representations because it is not compact. Now, I know that the "converse" part ...
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89 views

What is the relation between these two subgroups of a finite cyclic group?

Let $G$ be a finite cyclic group of order $n$ generated by $a$. If $k$, $m$ are integers such that gcd($n, m$) = gcd($n,k$), then what is the relationship between the subgroups generated by $a^k$ and ...
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Canonical Homomorphisms

Let $G$ be a group and let $N$ be a normal subgroup. Let $\pi\colon G \to G/N$ denote the canonical homomorphism. Prove that if $H$ is a subgroup of $G$ then $\pi(H) = \pi(HN)$. Then prove that if ...
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1answer
123 views

bijective correspondence involing homomorphsims to a direct product of groups?

Let $G$, $G'$ and $H$ be groups. Establish a bijective correspondence between homomorphisms $\Phi: H \to G \times G'$ from $H$ to the product group and pairs $(\varphi, \varphi')$ consisting of a ...
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197 views

Find quotient group $GL(n, \mathbb{C})/H$, where $H$ is a group of invertible matrices $GL(n, \mathbb{C})$ with $det \in \mathbb{R}$.

Find quotient (factor) group $GL(n, \mathbb{C})/H$, where $H$ is a group of invertible matrices $GL(n, \mathbb{C})$ with $det \in \mathbb{R}$. I suppose, the theorem that for any group homomorphism ...
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1answer
98 views

$p$-subgroup of $G$ with $p$ a prime.

If $H$ is a $p$-subgroup of a finite group $G$ then $[N_G(H) : H] ≡ [G:H]\operatorname{mod} p$. Proof: Let S be the set of left cosets of H in G. H acts on S by left translation, $h · aH = ...
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171 views

Find Orbit of element $m \in M, M = M(2, \mathbb{R})$ under action of group $G = GL(2, \mathbb{R})$ mapping $m$ to $g^{-1}mg$, $g \in G$.

Find Orbit of element $m \in M, M = M(2, \mathbb{R})$ under action of group $G = GL(2, \mathbb{R})$ mapping $m$ to $g^{-1}mg$, $g \in G$. The element $m$ is $ \left( \begin{matrix} 2 & 1 \\ 0 ...
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1answer
61 views

A subset of $Z(G)$ [duplicate]

If $H$ is a normal subgroup of $G$ and the $|H|$ is the smallest prime that divides $|G|$, then $H \subset Z(G)$. First I note that (1) $|G/C_G(H)|$ divides the $|Aut(H)|$, how can i prove this ...
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$G \subset Map(S,S)$, a subset closed under composition, has a unique projector

I'm having trouble showing that: For $G \subset Map(S,S)$ a subset closed under composition (i.e. for all $f,g \in G: f\circ g \in G$) such that $(G, \circ)$ is a group, $G$ has a unique projector ...
2
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Nilpotent Groups Characterization

Prove that G finite is nilpotent iff for every proper subgroup H of G, $N_G(H) ≠ H $. Since H is a proper subgroup of G, $Z_0(G) = {e} ⊆ H$, and G is nilpotent then there is a n such that ...