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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2x2 matrices and groups under multiplication

Let n be a positive integer. (a) Let G be a set of real 2x2 matrices $A$ such that the $detA$ is a rational number of the form $m/n^t$ where $m$ and $t$ are nonnegative integers and $m\ne0$. Is G a ...
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2answers
104 views

There are 2 groups of order 6 (up to isomorphism) [duplicate]

I need to show that there are only 2 groups of order 6 up to isomorphism. I did prove it, but the proof is quite cumbersome. I wonder if there is a very concise proof. My proof outline: Suppose $G$ ...
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1answer
42 views

Consider the homomorphism from A5 to Z_60. Show that the kernel is equal to A5.

I know that A5 is simple, and thus it has no non-trivial, proper subgroups. So the kernel must either be {e} or all of A5. But how do I show it's equal to A5/not equal to the trivial subgroup?
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In a finite permutation group where $|\mbox{fix}(g)| \le 2$ for $g \ne 1$. Properties of orbit of $O_p(G)G_{\alpha}$.

Let $G$ be a finite permutation group acting transitive and non-regular on $\Omega$ with $|\Omega| \ge 4$. Suppose further that every nontrivial element has at most two fixed points. Now let $\...
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3answers
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What's the angle of rotation of a product of two reflections?

Let $F_1$ and $F_2$ be two arbitrary reflections about two lines in $\mathbb R^2$. I've been trying to work out the angle of rotation of $R_1R_2$. To this end I drew pictures in which I reflect one ...
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3answers
69 views

Prove an abelian group of order six is cyclic

Suppose$G$ is an abelian group and $|G|=6$, prove that $G=\{e,g,g^2,...,g^5\}$ for some $g$ with $g^6=e$. My attempt: If $G$ consists only of elements of order 2, then $|G|=2^m$ for some $m$. $6 \...
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Do we count the identity element as an element of a finite group?

When talking about the order of a finite group, do we take the identity element into account? Here is a theorem I read from the book 'Algebra and Geometry' by Alan. F.Beardon: Let $G$ be a finite ...
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54 views

Geometric argument why rotation by 180 degrees commutes with reflections

I have no trouble determining the center of the dihedral group $D_n$ using an algebraic argument ($R_{180}$ is self-inverse). But I've been trying (without success) to find the geometric explanation ...
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27 views

how not both points can be on the polygon implies that every point on the polygon is uniquely determined by its distance from given two points?

I need help understanding the proof of lemma 2.1. in these notes here. The proof and lemma are the following: Lemma 2.1. Every point on a regular polygon is determined, among all points on the ...
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1answer
58 views

How would you show that a non-cyclic group of order 10 must contain an element of order 5?

How would you show that a non-cyclic group of order 10 must contain an element $r$ of order 5? Also, is every pair of groups homomorphic? Thanks
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68 views

$H = \{a^2 | a \in G\}$ is a subgroup of $G$

Let $G$ be a group. Suppose that the map $\Phi : G \to G$ given by $\Phi(a) = a^3$ is a group homomorphism, prove that For every $a, b \in G$, $baba = a^2b^2$. The subset $H = \{a^2| a \in G\}$ is a ...
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36 views

Infinite torsion group with K(G,1) of finite type

I am wondering whether any group $G$ that is torsion and has a $K(G,1)$ of finite type (i.e. there are finitely many cells in each dimension) is already finite. The condition of having a $K(G,1)$ of ...
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60 views

How do I find the commutator subgroup of $A_n$ for $n \ge 5$?

I understand it can't be trivial since $A_n$ is not abelian. Further, $A_n$ is simple so the only option remaing is that $[A_n, A_n] = A_n$. Is this correct?
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20 views

Infinite isometry in subset of Euclidean plane

Is there an example of a nonempty bounded subset of the Euclidean plane which has an infinite isometry group? Would the unit cube $[0,1]^n$ be an example?
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35 views

The preimage of a normal subgroup under a group homomorphism is normal

Let $\phi:G\to G'$ be a group homomorphism, and let $N'$ be a normal subgroup of $G'$. Show that $\phi^{-1}[N']$ is a normal subset of $G'$. My attempt: $\phi^{-1}[N']=\{g\in G:\phi(n)\in N'\}$ $$g' ...
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30 views

Upper and Lower Uni-triangular groups

Let $F$ be a field, and consider the $n\times n$ matrix groups $U$ and $L$ over $F$ as follows: $$ U=\begin{Bmatrix} \begin{bmatrix} 1 & * & \cdots & *\\ & 1 & \cdots & *\\ &...
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29 views

To prove $(a ∗ b^{−1} ∗ a)^3 = I$.

If $(G, ∗, I)$ is a group and $a, b ∈ G.$ satisfy $a^2=b^3=I$. Then I need to prove $(a ∗ b^{−1} ∗ a)^3 = I$. My Work:- $(a ∗ b^{−1} ∗ a)^3$ $a ∗ b^{−1} ∗ a * a ∗ b^{−1} ∗ a * a ∗ b^{−1} ∗ a$ $...
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36 views

Every subgroup of a group has left cosets (True or False)

This is true, because one can always multiply elements of $G$ with elements of $G$ (in particular, the ones that are contained in $H$), and that is a non-empty subset of $G$, because G is a group. Is ...
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49 views

When is the $k/n$ representation of $D_n$ irreducible, and why?

The $k/n$ representation of the Dihedral group of order $2n$ in $GL(2,\mathbb{C})$ is induced by mapping the rotation element of $D_n$ to the Rotation Matrix $R(\frac{2\pi k}{n})$, and the reflection ...
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29 views

Showing that intersection of cyclic groups is identity

Let $|a|=10, |b|=21$. Show that $<a>\cap<b>=${$e$} I think that $|a|=10$ implies there are subgroups of the cyclic group of order $2,5$; $|b|=21$ implies that there are subgroups of order ...
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What are the algebraic structures of the wallpaper groups?

The wallpaper groups are discrete groups of affine motions in the plane that contain two linearly independent translations. Cf. https://en.wikipedia.org/wiki/Wallpaper_group Some of them have very ...
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Does the additive closure of a set always have additivity?

If I have some set X and define the set XC as the additive closure of X, is it possible that XC does not have additivity? Additivity here is defined as follows: for every $a,b \in X$ ,$ a+b \in XC$ ...
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33 views

Rank of the Homomorphic image of a finitely generated abelian group

I have been trying to find a relatively simple proof of the following result: let $f$ be a group homomorphism such that $f: G \to K$ where $G$ and $K$ are groups. Then, $rank(f(G))\leq rank(G)$. ...
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4answers
318 views

Prove that a group is cyclic [closed]

$G$ is abelian of order $35$. and for all $x\in G$, $x^{35}=e$. I need to show that $G$ is cyclic. This seems perfectly obvious but I dont know how to write the proof. Help would be appreciated! ...
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Completing the proof using the fundamental theorem of cyclic groups

$G$ is abelian; $H=${$g\in G,|g| \text { divides } 12$}. Prove that $H$ is a subgroup of $G$. My idea: if the order of $g$ divides 12, then the order of the group $G$ must be some multiple of $12$. ...
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1answer
35 views

If $K$ is a normal subgroup of $G$, is it true that $G$ is isomorphic to the direct product $K \times (G / K)$?

Let $K$ be a normal subgroup of the group $G$. Is it true that the direct product of $K$ by $G/K$ is isomorphic to $G$? Which isomorphism can we define from $K\times G/K$ to $G$?
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132 views

Number of non-isomorphic group of order 8?

Number of non-isomorphic group of order 8 ? 2 3 4 5 I have no clue how to solve this. It's a group of order $p^3$ so is there any general way to calculate the number of non-isomorphic groups of ...
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About centralizers of involutions in finite simple groups

I read that for the classification of the finite simple groups the centralizers of involutions plays a crucial role. This has to to with the Brauer-Fowler-Theorem, which in one formulation could be ...
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How can you tell whether two groups are homomorphic/isomorphic? [closed]

Suppose you have two groups, $G$ and $H$. I've been taught the following definitions: "$G$ is homomorphic to $H$ iff there exists some function $\theta$ which gives the mapping $\theta : G \...
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Express an abelian group given as finite generators and their relations as a direct sum of cyclic groups and find corresponding generators.

According to page 158 of Dummit and Foote's Abstract Algebra (3rd edition): Theorem. (Fundamental Theorem of Finitely Generated Abelian Groups) Let $G$ be a finitely generated abelian group. Then ...
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Is the group G of rotations preserving a regular tetrahedron cyclic? [closed]

Also, am I correct in thinking that the order of this group is $|G|=12$? Many thanks
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60 views

Commutator subgroup is the minimal normal subgroup such that quotient group is abelian [duplicate]

I was recently asked this in Abstract Algebra class on group theory: Let G be a group and G' its commutator subgroup (i.e. its minimal subgroup containing all commutators $ [x,y] = xyx^{-1}y^{-1} $...
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2answers
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Is $G/H$ cyclic when $G=S_5 \times \mathbb{Z}$ and $H=A_5 \times 97\mathbb{Z}$?

I think it is not because $G$ is non-abelian (because $S_5$ is not) and therefore the quotient cannot be abelian. And then since it is not abelian, it couldn't have been cyclic to begin with.
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1answer
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Let $ H \lt G $ such that $ [a,b] \subset H $. Then H is a normal subgroup of G

Let $ H \lt G $ such that $ [a,b] \subset H .$ Then $H$ is a normal subgroup of G . I am not sure if what i did was a good way to proceed. Proof. Let $ a,b \in G, $ and $ aba^{-1}b^{-1} \in G' $ (...
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Showing that the conjugates of a proper subgroup do not cover the group.

I am trying to figure out the following. Suppose $G$ is a group and is finite; let $H$ be a proper subgroup. Show that the conjugates of $H$ do not cover $G$ (that is, there is some $g \in G$ which ...
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1answer
31 views

Is this the right way to find how many distinct subgroups $\mathbb{Z}_{12}$ has?

I'm considering $\mathbb{Z}_{12}=\{0,1,2,3,4,5,6,7,8,9,10,11\}$, with the operation of addition modulo 12. To find the distinct subgroups of this, I assume that I would attempt to find each cyclic ...
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125 views

Determine whether or not φ is a homomorphism:

Let φ : Z6 -> Z2 be given by φ (x) = the remainder of x when divided by 2, as in the division algorithm. I know that this is ...
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16 views

On the image of $\pi$-Hall subgroup in $G / G'$.

Let $G$ be a finite group and $\pi$ a non-empty set of primes and let $P$ be a $\pi$-Hall subgroup of $G$. Then we have $PG' / G' = O_{\pi}(G/G')$, and so as $G/G'$ is abelian $$ G / G' = PG'/G' \...
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1answer
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Show that if Z(N) = {e} and Z(G/N) = {e}, then Z(G) = {e}.

Let G be a group and let N be a normal subgroup of G. Show that if Z(N) = {e} and Z(G/N) = {e}, then Z(G) = {e}. Z(G) is the set of elements in G that commute with all of G. I have no idea what to ...
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Connecting homomorphism is Bockstein operation, construction of natural long exact sequence.

Let $0 \to \pi \overset{f}{\to} \rho \overset{g}{\to} \sigma \to 0$ be an exact sequence of Abelian groups and let $C$ be a chain complex of flat Abelian groups. Write $H_*(C; \pi) = H_*(C \otimes \pi)...
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41 views

Group Theory Cyclic Generators Proof

Suppose $a$ is a power of $b$, say $a=b^k$, then $b$ is equal to a power of $a$ if and only if $\langle a\rangle=\langle b\rangle$. I am not even sure how to really start this. I want to say ...
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103 views

How do you find the generator for an additive group?

I know that for a multiplicative group $G$, that $x \in G$ is a generator of $G$ iff all elements in $G$ are $x^{n}, n \in \mathbb{Z}$. Is this the same for an additive group? If not, why? What about ...
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46 views

Show the following is a group $a * b=a+b+ab$

Let $S=\mathbb{R}\setminus \{-1\}$. Define $*$ on $S$ by $a*b=a+b+ab$. This feels fairly straightforward but it is just the details of which approach is valid that is throwing me. My attempt: We ...
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17 views

Concrete applications of theory of classes of groups

I'm reading chapter II of the book "finite soluble groups" by Doerk-Hawkes, and I wonder what are some applications of the theory of classes of groups and particularly of the theory of formations.
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Listing the elements of the subgroup

I have a minor calculation mistake here. I am asked to list the elements of subgroups $\langle 3\rangle$ in $U(20)$ So I write $U(20)= \{1,3,7,9,11,13,17,19\}$ $\langle 3\rangle = \{3,6,9,12,15,18,...
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If all Sylow subgroups of $G$ intersect in Sylow subgroups of solvable subgroup $H$, then $H$ is subnormal

Let $H \le G$ be a solvable subgroup of the finite group $G$ such that for each prime $p$ and each Sylow $p$-subgroup $S$ of $G$ we have $$ S \cap H \in \mbox{Syl}_p(H). $$ Then $H$ is subnormal ...
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22 views

Bijective correspondence between subdirect product of two groups and fiber product

I want to show, knowing the Goursat's theorem, that given two groups $G, G'$ and a subgroup $H \subset G\times G'$, the projections $p_1: H \rightarrow G $ and $p_2:H \rightarrow G' $ are surjective (...
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Find all subgroups of $\mathbb{Z}_{12}\times \mathbb{Z}_{4}\times \mathbb{Z}_{15}$ of order $9$

Find all subgroups of $\mathbb{Z}_{12} \times \mathbb{Z}_{4} \times \mathbb{Z}_{15}$ of order $9$. I am able to find a subgroup, but I am not sure how to find all of them
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3answers
87 views

Show that $a^{m} a^{n} = a^{m+n}$

Prove that if $G$ is a group and $a\in G$, then we have $\forall m,n\in\mathbb{Z} $ that $$a^{m} a^{n} = a^{m+n}.$$ I've proved the case when $m,n>0$ but I'm stuck on how to prove the case when ...
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Subgroup $Q$ of $M\times M'$ is not a direct product of subgroups of $M$ & $M'$, do $M$ and $M'$ share no simple subgroups?

Let $M$ and $M'$ be groups. Let $M\times M'$ be a direct product. If a subgroup $Q$ of $M\times M'$ is not a direct product of subgroups of $M$ and $M'$, in other words, $Q\neq \{(m,m') \mid m\in P\le ...