# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### $H = \{a^2 | a \in G\}$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...