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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Consequence of Lemma: If G is abelian with exponent n, then $|G|\big\vert n^m$ for some $m\in N$

Lemma: If G is abelian with exponent n, then $|G|\big\vert n^m$ for some $m\in N$. Theorem to be proved: Suppose G is finite abelian and group of order m, let p be a prime number dividing m. Then G ...
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1) Show that $(\Bbb Z[\sqrt2]^*, .)$ is infinite.

1) Show that $(\Bbb Z[\sqrt2]^*, .)$ is infinite. 2) Classify $(\Bbb Z[\sqrt2]^*, .)$, where $\Bbb Z[\sqrt2]^*$ is the group of units of $\Bbb Z[\sqrt2]$ What I have done so far that for $a+b\sqrt2$ ...
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Every Two Element in A Coset

For every $a,b \in G$, $a,b \in cH$ for some maximal subgroup $H$ of $G$ and some $c \in G$. For what groups is the following property true? I know its true for $\mathbb{Z_m} \times ...
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Prove that $C_H(K)=N_H(K)$.

Let $H$ & $K$ be groups. Let $\phi$ be a homomorphism from K into $Aut(H)$. Identify $H$ & $K$ as subgroups of $G=H\rtimes_{\phi}K$. Prove that $C_H(K)=N_H(K)$.
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How to find orbits and isoropy group?

About this problem ${a}$, I am wondering if there are 5 orbits in $A$? The 5 orbits separately contain elements which 3 are all the same, 2 of 3 are the same and all 3 are different? I am confused ...
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Group $G$ such that there is a proper subgroup containing every other proper subgroup of $G$

Characterize all the groups $G$with the following property: There is a proper subgroup $H$ of $G$ such that $\forall S$ proper subgroup of $G$, $S \subset H$. I am pretty lost with this exercise. If ...
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Integral domain and ordered ring

I know that every ordered ring is an integral domain. Is it true that every ring that is an integral domain is also ordered? By ordered I mean for any $a \in R$, either $a = 0$ or $ a > 0$ or $a ...
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Prove that $n_p(N)$ divides $n_p(G)$

Let $N$ be a normal subgroup of $G$ where $G$ is finite group, then we have to prove $n_p(N)$ divides $n_p(G)$ ( here $n_p(G)$ means number of sylow $p$-subgroups of $G$) I was able to prove that ...
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Does the binary operation $m ⋆ n = m^n$ on $\mathbb N$ have a neutral element?

Does the binary operation $\,m ⋆ n = m^n\,$ on $\,\mathbb N\,$ have a neutral element? I said yes, and it is $\,e=1\,$ because $\,m ⋆ e = m^e = m^1 = m,\;$ but apparently that is wrong.
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144 views

Group of order $|G|=pqr$, $p,q,r$ primes has a normal subgroup of order

Let $p,q,r$ be positive primes, $p<q<r$, and let $G$ be a group with $|G|=pqr$. Show that there exists a normal subgroup $H$ of $G$ of order $qr$. I've seen this post Groups of order $pqr$ and ...
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Give a $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$

Consider $SL_{2}(\Bbb Z_p)$ if q & p be two primes, $p>q$. Give an example of a subgroup $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$ when i) $q|(p-1)$ ii) $q|(p+1)$
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Conjugacy classes in non-solvable group

Suppose $A$ is an arbitrary subset of group $G$ and $K_G(A)$ be the number of $G$-conjugacy classes contained in $A$. Also suppose $G$ non-solvable group, $N\unlhd G$, $G/N$ is abelian, $|G/N|=6$ and ...
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48 views

Elementary abelian $p$-group

How can we show that if $N$ is abelian and $C_G(N)=N$, then $N$ is an elementary abelian $p$-group for some prime $p$?
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51 views

Classification of groups of order $p^2q$

I have done the classification of groups of order $p^2q$, where $p>q$. $p,q$ odd distinct prime If $P\in Syl_{p}(G)$ & $Q\in Syl_{q}(G)$ then Case $1$: $P=\Bbb Z_{p^2}$. I have done. But my ...
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Number of conjugacy classes in a finite non-abelian simple group

Can we say that every finite non-abelian simple group has at least 4 non-identity conjugacy classes?
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Separating invariant of a group action

Let $G = (\mathbb{R},+)$ be a group, $M = \mathbb{R}^2$, $$\omega \colon \mathbb{R}\times\mathbb{R}^2\to \mathbb{R}^2, \quad \left(t, (x,y)\right) \mapsto(x+t,y-2t)$$ and $$\iota \colon\mathbb{R}^2 ...
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186 views

Operations on power set

Which kind of operations on a power set leads to a group/monoid? Known to me are: - intersection - union - symmetric difference - complex product of a group Ate there some more examples?
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129 views

Isomorphism classes of abelian groups of certain order

working on a practice question about finite abelian groups and just want to see if I am on the right track: Let $H = <(123)(4567),(8\space 9)(10\space 11),(8\space 11)(9\space 10) > \space ...
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Isomorphism group between $\mathbb R$ and one of it's propre subgroups.

Is there a subgroup $H$ of $(\mathbb R,+)$ such that $H \neq \mathbb R$ and $H \simeq \mathbb R$ as groups?
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150 views

Why is the dihedral group closed under composition?!

I've been obsessing over this all day now. I understand associativity, presence of inverse elements and identity, but I don't get why a composition of a reflection with a rotation or other reflexions ...
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56 views

Semi direct product

Prove that (i) $GL_n(R)= \coprod_{w\in S_n} UwB$ where $w \in S_n$ is a permutation matrix. and $U$ is a subgroup of $GL_n(R)$ consisting of upper triangular matrices with diagonal entries $1$ and ...
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Subgroups and justification

Which of these subsets are subgroups of the given group and justify your answer. The group $R^+$ of postive reals under multiplication. The subset $H=(3n|$ $n\in Z^+)$. The group of nonzero ...
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107 views

Are all torsion groups finite groups?

Are all torsion groups finite groups? I've been trying to find a counter example, but have had no luck so far. Can anyone throw me one, or give me an idea to prove this?
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How often are Galois groups equal to $S_n$?

Let $\mathbb{Z}[x]_n$ be the set of polynomials in $\mathbb{Z}[x]$ of degree at most $n$. Then, consider some sensible increasing filtration $$A_0 \subset A_1 \subset A_2 \subset \cdots$$ of ...
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78 views

What does the notation $H=\{ a | a^2=e \}$ mean?

What is the meaning of notation $H=\{ a | a^2=e \}$? Is it the same as $H=\{a,a^2=e\}$? (Here $a$ is an element of some group, with identity $e$.)
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59 views

Groups of Order 2 with subgroups

Let G be an abelian group and $a,b\in G$ be two distinct elements with a and b or order $2$. Show that $H=\{e,a,b,ab\}$ forms a subgroup and write out its multiplication table. Justify why all the ...
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77 views

On infinite groups with unique minimal subgroup

Let $\operatorname{Sub}(G)$ be the lattice of all subgroups of an infinite (abelian) group $G$. If $\operatorname{Sub}(G)\setminus\{\{1\}\}$ has a minimum element which is a cyclic subgroup of prime ...
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Presentation of a group: Show that $\langle a|a^2\rangle =\{1,a\}$.

Example: $\langle a|a^2\rangle=\{1,a\}$. After reading the definition of presentation of a group, I find myself cannot understand the above example given. I don't know which part of the definition I ...
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When can a group be decomposed into a direct product of smaller groups?

Is there any general condition that a group must satisfy in order to be decomposable into a direct sum or product of smaller groups? And what happens if one replaces 'direct product' with semi-direct ...
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Definition of $p$-supersolvable groups

"A group is said to be $p$-supersolvable if it is $p$-solvable and every $p$-chief factor has order $p$." Is this an accurate definition of $p$-supersolvable groups?
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Solvability of a group

What is the intuition behind the solvable groups? It is defined by composition series. Is there any intuitive way to understand it?
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On a finite group with unique minimal subgroup

EDIT: Let $G$ be a finite group with a unique minimal subgroup. Then we know that $G$ is a $p$-group. Let $p\neq2.$ Is it true that every subgroup $H\le G$ and every quotient $G/N$ have a unique ...
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Counting the number of elements in a double coset

Let $G$ denote the groups of $n\times n$ invertible matrices and $H$ be the subgroup of invertible upper triangular matrices. For $n=2$, by row reduction, or equivalently LU decomposition, it is ...
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Sylow subgroups of a non abelian group $G$ with $|G|=21$ and $|G|=39$

I am trying to solve the following exercise: ¿How many Sylow subgroups has a non abelian group $G$ of order $21$ and $39$ respectively. I could do the following: a) $|G|=21=3\cdot 7$. I'll call ...
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62 views

Definition of “the same”

Given a subgroup H of a group G such that $g^{-1}hg \in H$ for all $g \in G$ and all $h \in H$, I need to show that every left coset gH is the same as the right coset Hg. In the context of this ...
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81 views

Classification of subgroup of $\mathbb{Z}^n$.

Fix an integer $n\geq 2$, can we list all subgroups of $\mathbb{Z}^n$?
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Cycles of odd length: $\alpha^2=\beta^2 \implies \alpha=\beta$

Let $\alpha$ and $\beta$ be cycles of odd length (not disjoint). Prove that if $\alpha^2=\beta^2$, then $\alpha=\beta$. I need advice on how to approach this. I recognized that $\alpha,\beta$ are ...
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Groups and U27 double check

This is just a quick question. The Group U$_{27}$=$(1,2,3,5,7,11,13,17,19,23)$ right? Or am I just very wrong here?
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Proofread my work: Expressing generators of a cyclic group

The following question comes from Serge Lang's Undergraduate Algebra(pg. 26, 3rd edition). I just learnt the concept of groups and subgroups and I spent an hour or so on tackling part (b) of this ...
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Conditions for a finitely generated group with finite ordered generators

What are the conditions for a finitely generated group $G$ with finite ordered generators say $a_1, a_2,...,a_n$ to be finite? Note:I know that if $G$ is abelian, then it is finite. Are there any ...
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76 views

Prove that if $g^{n} \in H$, n divides |g|

Let H be a subgroup of a finite group G. Suppose that g belongs to G and n is the smallest positive integer such that $g^{n} \in H$. Prove that n divides |g|. I couldn't get anywhere with this ...
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Non-isomorphic groups

How to prove that $Z/2\times Z/2$ and $ Z/4$ are not isomorphic? I think that $Z/2\times Z/2$ is not cyclic. Hence $Z/2\times Z/2$ and $ Z/4$ are not isomorphic. Thank you.
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Showing if $G$ is a group then the centralizer of an element is a subgroup

Given: Let $G$ be an arbitrary group, and let $a\in G$. The centralizer of $a$ is defined as $$C(a)=\{x\in G: xa=ax\}.$$ Question: Show that if $G$ is a group and $a\in G$, then $C(a)$ is a subgroup ...
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Restricting a representation to a subgroup

This little factoid from algebra quals stumped me: Let $G$ be a finite group and $H \triangleleft G$ an index $2$ subgroup. If we take an irreducible complex representation $V$ of $G$ and restrict it ...
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Showing associativity holds over n elements

Say we have a set $X$, with an associative binary operator $*$. How can we show that for any string $x_1 x_2 \ldots x_n$, when we insert brackets or the operation $*$, we will always get the same ...
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Equivalent actions with normalizing element

Suppose $x \in G$, $G$ a group, $H$ a subgroup of G, and $G$ acts on a set $X$. Then if $x$ normalizes $H$, I know that we can say that $x$ permutes the orbits of $H$ on $X$. How can we prove that ...
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70 views

Normal subgroup and index problem

Let $G$ be a group and let $N$ be a normal subgroup of $G$ of finite index. Show that if $H$ is a finite subgroup of $G$ whose order is coprime with $[G:N]$, then $H$ is a subgroup of $N$. I don't ...
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algorithm to find the order of $a$ in $(\mathbb{Z}/n\mathbb{Z})^*$ where $n$ is not prime.

algorithm to find the order of $a$ in $(\mathbb{Z}/n\mathbb{Z})^*$ where $n$ is not prime. Now I know a naive algorithm where you just keep multiplying $a$ by itself until you find it equals $1 \mod ...
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Index of center $Z(G)$ is finite implies the number of elements of conjugacy class is finite

Exercise Let $G$ be a group such that its center $Z(G)$ has finite index in $G$. Show that every conjugation class has finite elements. I don't know how to attack the problem. I thought the ...
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Short exact sequence with binary tetrahedral group does not split

The following is a short exact sequence, where $T$ is the binary tetrahedral group (equivalently the Hurwitz units), and $Q$ is the quotient of $T$ by $\mathbb{Z}/2$. $1 \rightarrow \mathbb{Z}/2 ...