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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30 views

why if $\frac{G} {Z(G)} \cong Z_P. Z_P$ then $G' \cong Z_P$? [on hold]

why if $\frac{G} {Z(G)} \cong Z_P. Z_P$ then $G' \cong Z_P$? can you help me?
10
votes
2answers
263 views

What's the automorphism group of the real and complex numbers and the quaternions?

According to Wikipedia the automorphism group of the octonions is the exceptional group $G_2$. Are there analogous groups for the real numbers, the complex numbers and the quaternions?
1
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2answers
30 views

Triangle group $(\beta,\beta,\gamma)$ is a subgroup of the triangle group $(2,\beta, 2\gamma)$.

Let $1/\alpha+1/\beta+1/\gamma<1$, and let us consider the triangle group $(\alpha,\beta,\gamma)$, i.e. the subgroup of $\mathbb{P}\mathrm{SL}(2,\mathbb{R})$ induced by the hyperbolic triangle ...
2
votes
1answer
34 views

$G$ is an infinite abelian group such that $G \cong H$ for every non trivial subgroup $H$ of $G$ , then is $G$ cyclic?

If $G$ is an infinite abelian group such that $G \cong H$ for every non trivial subgroup $H$ of $G$ , then is $G$ cyclic , or equivalently asking , then is $[G:H]$ finite for every non trivial ...
2
votes
1answer
65 views

$G$ be an infinite group such that $|Aut (G)|=2$ , then is $G$ cyclic ?

Let $G$ be an infinite group such that $|Aut (G)|=2$ , then is $G$ cyclic ? Since $Aut(G)$ is cyclic here , I know that $G$ is abelian , but this is as far as I can get . Please help . Thanks in ...
5
votes
1answer
43 views

Chain Rule of Calculus as a Group Property?

I read that the chain rule and inverse function theorem are expressions of the group property of successive non-singular transformations. How do you say this more formally? My guess is that we are ...
0
votes
0answers
20 views

Proving a conjugation map is an Inner automorphism of a group

Definition: The map $i_{g}:G\rightarrow G$ $h\mapsto g^{-1}hg$ Lemma: $i_{g}$ is an Automorphism of G called an Inner Automorphism. My attempt to prove this is as follows: ...
-1
votes
0answers
29 views

Show that $B$ is an infinite group. [on hold]

So $B$ denotes the group with presentation $\langle s,t\mid stst^{-1}s^{-1}t^{-1}\rangle$. What is a good way to prove this? Thanks a lot.
1
vote
1answer
34 views

number of generators of a semi direct product

Let $G$ be a finite group. Let $g(G)$ be the minimum set of elements of $G$ required to generate the whole group. Suppose that $G= H \rtimes K$ is a semi direct product of two finitely generated ...
2
votes
0answers
67 views

Prove neither $F_4$ nor $F_5$ is isomorphic to $\mathbb{Z}^4$?

Prove neither $F_4$ nor $F_5$ is isomorphic to $\mathbb{Z}^4$. ($F_4,F_5$ are free groups.) My first thought is to show that $F_4$ and $F_5$ are not free abelian groups and thus cannot be ...
0
votes
0answers
34 views

Find the orbit of $x$

Let $G=D_{2n}$ and let $g\dot{}x=gxg^{-1}$ define and action of $G$ on itself. Find the orbit and stabilizer if $x=a.$ Note: Here $a$ denotes a rotation through $\frac{2\pi}{n}$ and $b$ denotes a ...
0
votes
1answer
26 views

Orbit and stabilizer

Let $G$ be a group and let $g \dot{}x=gxg^{-1}$ for all $g,x\in G.$ Find the orbit and stabilizer when $x=e.$ Orbit: $$G\dot{}x=\{g\dot{}x \ \colon g\in G\}=\{g\dot{}e \ \colon g\in G\}=G$$ ...
0
votes
2answers
39 views

H prime order, normal subgroup of group G. Prove H in center Z(G).

I am looking at the following question: "Let H be a normal subgroup of prime order p in a finite group G. Suppose that p is the smallest prime that divides the order of G. Prove that H is in the ...
-2
votes
1answer
29 views

Prove that a semigroup which satisfies a certain conditions is a group [on hold]

This is an exercise from "Abstract Algebra" by P.A.Grillet (p.12, ex.2). Let $S$ be a semigroup (that is, a set with an associative binary operation) in which there is a left identity element ...
0
votes
0answers
17 views

Prove that if $H$ is an abelian subgroup of a group $G$ then $\langle H, Z(G)\rangle$ is abelian.

Prove that if $H$ is an abelian subgroup of a group G then $\langle H, Z(G)\rangle$ is abelian. So I started out trying to show that $H$ is a subset of $Z(G)$ but then I realised that $Z(G)$ is the ...
2
votes
0answers
19 views

about minimal non-nilpotent groups

Newman and Wiegold have studied the AN-groups i.e. the locally nilpotent groups which are not nilpotent but every proper subgroup is nilpotent. I was asking why the notion locally nilpotent was added ...
0
votes
4answers
59 views

Why is the 'Law of Cancellation' for groups only an implication?

It is easy to see that for a Group $G$ and $a,b \in G$ $ab = ac \Rightarrow b = c$ (See also here) But what is about the other direction? That is: $b = c \Rightarrow ab = ac$ Does this ...
1
vote
2answers
22 views

Definition of subgroup of abelian group $G$ generated by subset $A$

In my book I have the following definition for subgroups of a group $G$ generated by $A$, a subset of G: $$\langle A\rangle=\{x_1^{\epsilon_1}x_2^{\epsilon_2}...x_n^{\epsilon_n}\mid x_i\in ...
0
votes
1answer
29 views

When does the union of distinct cosets equal the group?

I have an example where it's claimed that the union of distinct cosets of group $G$ spanned by $a$ and $b$, $a≠b$, equals the entire group $G$. Is this a general property?
3
votes
1answer
50 views

Intuitive explanation for the connection between Lie Groups and projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$

In this post John Baez states that the classical simple Lie groups "arise naturally as symmetry groups of projective spaces over $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$". Is there some ...
0
votes
1answer
59 views

When is this cyclic representation irreducible?

Let $G$ be a finite group, and let $(\rho, W)$ be a representation of $G$ on $W$. We assume that $W = \bigoplus_i W_i$ is a direct sum of equivalent irreducible representations $W_i$. There are many ...
0
votes
0answers
24 views

Cardinality of stabilizer in $PSL(2,\mathbb{Z}_7)$

Let $v$ be a non-zero vector in $\mathbb{Z}_7^2$ (up to scaling), and let P be the stabilizer of $v$ (up to scaling) in the Projective special linear group $PSL(2,\mathbb{Z}_7)$. Find the Cardinality ...
0
votes
1answer
34 views

Every proper maximal subgroup of a $p$-group $P$ is normal and has index $p$.

Every proper maximal subgroup of a $p$-group $P$ is normal and has index $p$. I tried to search online by I can't get a complete proof. Take $M$ to be maximal and $Z$ to be central subgroup of order ...
-1
votes
1answer
31 views

Factoring polynomials in $\Bbb Z_n$

a). Factor $f(x) = x^3 + 4x^2 + 5x + 2$ completely over $\Bbb Z_7$. b). Give two different factorizations of $x^2 + x + 8$ in $\Bbb Z_{10}[x]$. I have found the zeros of both of these but I am ...
0
votes
1answer
56 views

Find all the zeros of $f(x) = x^3 + 3x + 5$ in $\Bbb Z_7$

Find all the zeros of $f(x) = x^3 + 3x + 5$ in $\Bbb Z_7$. I've tried factoring this into multiple forms but I can't seem to find an easy way to find the $x'$s for $x^3 + 3x + 5 = 0$. Any hints or ...
0
votes
0answers
25 views

topology notation $G \backslash X$

Suppose there is a group $G$ acting on space $X$. What does the following notation mean? $$G \backslash X$$ Thank you!
9
votes
1answer
77 views

Finding the kernel of an epimorphism onto $S_3$?

Let $\Lambda$ denote the group with presentation $\langle a,b \mid abab^{-1}a^{-1}b^{-1}\rangle$. We define the following epimorphism from $\Lambda$ onto $S_3$: $\theta: \Lambda(a,b) ...
0
votes
0answers
21 views

Partitioning the set of mappings.

The following is first two steps of an algorithm given from a research paper. I understood the first step. But please explain the second step: what does mean " Rearrange the partition according to ...
-3
votes
1answer
51 views

Isomorphic or not: two infinite groups [on hold]

The groups $(\mathbb{C}\setminus \left \{ 0 \right \},\cdot )$ and $(\mathbb{R},+)$ are not isomorphic. So I was not clear whether this statement is true or not.
3
votes
1answer
37 views

$\{(1), (12)(34), (13)(24), (14)(23)\}$ is the only non-cyclic proper subgroup of $A_4$?

What can I use to display the following: $\{(1), (12)(34), (13)(24), (14)(23)\}$ is the only non-cyclic proper subgroup of $A_4$. What I've started to do: list all the elements of $A_4$ and finding ...
4
votes
1answer
64 views

Prove that if Aut($G$) is the trivial group, then so is $G$? [duplicate]

Let $G$ be a finitely generated group. Show that if Aut($G$) is the trivial group, then so is $G$. I know that if Aut($G$) is the trivial group then $G$ must be abelian but I'm not sure how to ...
0
votes
1answer
70 views

What is an order of an element of a partition"?

I'm reading a paper, in which the set of all 3^3 mappings from {0,1,2} to itself (for instance {001,020,110,121,122}, {002,010,112,011}, {0,1,2}, ...) is partitioned, after which is written two ...
4
votes
1answer
38 views

Constructing well-defined epimorphism onto $S_3$?

Let $\Lambda$ denote the group with presentation $\langle a,b \mid abab^{-1}a^{-1}b^{-1}\rangle$. Construct an epimorphism from $\Lambda$ onto $S_3$, making sure to check that the function is ...
0
votes
0answers
17 views

$U$ subset of $S$, $G$ acts transitively on $S$, show that the subsets $gU$ cover $S$ evenly

I have a finite set $S$ on which a group $G$ acts transitively. Now, I let $U$ be a subset of $S$. I want to show that the subsets $gU$ cover $S$ evenly, meaning every element of $S$ is in the same ...
0
votes
2answers
31 views

$f(x) \in \mathbb{R}[x]$. Prove that if $z = a + bi$ is a zero of $f(x)$ then $z = a − bi$ is also a zero of $f(x)$.

Let $f(x) \in \mathbb{R}[x]$. Prove that if $z = a + bi$ is a zero of $f(x)$ then $z = a − bi$ is also a zero of $f(x)$. I'm learning about polynomial rings but my book and my instructor never ...
1
vote
1answer
22 views

The minimal group with Fitting length three has $p$ section in middle?

Let $G$ be a group with Fitting lengt $3$ i.e $$e< F_1< F_2 < F_3=G$$ and $F(G)=F_1$ and $\bar {F_2}=F(G/F_1)$. Assume that for every proper characteristic subgroup $K$ of $G$, Fitting ...
2
votes
1answer
46 views

Is this group given presentation isomorphic to $\mathbb{Z}_2$, and why?

We have the group $G = \langle a, \,b \; | \; a^2b^2ab^{-1}, \, a^3b^4a^{-2}b^{-3}\rangle$. Obviously $G/G' = \mathbb{Z}_2$. Is it true that $G = \mathbb{Z}_2$ or equivalently that $G'=1$? It is ...
0
votes
1answer
54 views

Don't understand the proof of Artin's “Algebra” Ed 1, Prop 5-8.4

I'm reading Artin's Algebra, Edition 1. In Chapter 5 there's proposition (8.4): Let $c_g$ denote conjugation by $g$, the map $c_g(x) = gxg^{-1}$. The map $f: S_3 \rightarrow Aut(S_3)$ from the ...
2
votes
1answer
59 views

Can we find a non central element of order 2 in a specific 2-group?

Let $G$ be a non-abelian group of order $2^5$ and center $Z(G)$ is non cyclic. Can we always find an element $x\not\in Z(G)$ of order $2$ if for any pair of elements $a$ and $b$ of $Z(G)$ of order ...
1
vote
0answers
38 views

Proof of Cayley's Theorem

This question relates to the link: Cayley's theorem The way I reasoned in showing the map T is a Homomorphism is as follows: Definition: A Homomorphism $\phi: \left ( G,\ast \right ) ...
1
vote
1answer
38 views

Can every group be obtained from a choice of Sylow subgroup for every prime divisor?

The question is almost clear from title: If $G$ is a finite group of order $p_1^{n_1}\cdots p_r^{n_r}$ then is it always possible to choose one Sylow subgroup for every prime divisor of $|G|$ ...
0
votes
1answer
43 views

Is the size of the conjugacy class of a given element in a compact Lie group always finite?

Let $G$ be a compact Lie group and $g\in G$ be any given element in it. Consider the conjugacy class of $g$ in $G$, denoted by $[g]=\{hgh^{-1}:h\in G\}$. Our question is that: Could you find a ...
-1
votes
0answers
11 views

Example of lattice of subgroups of quotient group [closed]

I've studied a theorem that explains what is the lattice of subgroups of a quotient group. The result is the following: Given a group G and a normal subgroup N if we denote by Sub(G) the lattice ...
0
votes
1answer
15 views

Is there a direct sum decomposition of the tensor product of two representations of two group elements?

I know that I can decompose $\rho_a(g) \otimes \rho_b(g)$ into $U^\dagger \left[ \rho_c(g) \oplus \rho_d(g) \right] U$. Is there a similar way to decompose $\rho_a(g_1) \otimes \rho_b(g_2)$ into ...
3
votes
1answer
34 views

Show the intersection of six subgroups of order $24$ is normal in $G$? [duplicate]

Let $G$ be a group with exactly six subgroups of order $24$. Show that the intersection of these six subgroups is normal in $G$. My thought is that if we can show these six subgroups are normal ...
1
vote
1answer
46 views

How many distinct patterns exist for a 5x5 grid by filling 3 colors?

Using 3 colors to fill in a $5\times5$ grid (you don't have to use all colors), then how many distinct patterns exist? The "distinct" means we have to consider the symmetry. Any effective approach is ...
2
votes
1answer
51 views

Show that if a group $G$ has odd order, any subgroup $H$ of index 3 in $G$ is normal in $G$? [duplicate]

Show that if a group $G$ has odd order, any subgroup $H$ of index 3 in $G$ is normal in $G$. I think this is equivalent to the following: Let $H$ and $K$ be subgroups of a group $G$, with $K ...
5
votes
1answer
44 views

Group with exactly six Sylow 5-subgroups?

Give an example of a group with exactly six Sylow 5-subgroups. I think $A_5$ works because it has 6 subgroups of order 5: $\langle(12345)\rangle,\langle(12354)\rangle, \langle(12435)\rangle, ...
5
votes
1answer
75 views

Example of a non-abelian quotient of a non-abelian finite group?

Give an example of a non-abelian quotient of a non-abelian finite group. This should be fairly simple but I am drawing a blank. I can think of plenty of non-abelian, finite groups but no ...
1
vote
1answer
59 views

What is the next term in the jumping champions?

I am interested in the jumping champions of the number of groups of order $n$, where $n$ is cubefree. After calling LoadPackage("cubefree"); GAP gives the ...