The study of symmetry: groups, subgroups, homomorphisms, and group actions.

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2
votes
1answer
28 views

Show that the free product of countably many countable groups is countable.

The question I am struggling with is as follows: Suppose that $\{G_\alpha\}$ is a countable collection of countable groups. Show that $\ast_{\alpha}G_\alpha$ is countable. The definition of ...
1
vote
1answer
20 views

Looking for examples of finite loops and monoids

I am looking for examples of (small) finite loops and monoids that are not groups for demonstrating what happens if you omit some of the group axioms. Does anyone know some ressources for this? I ...
0
votes
1answer
21 views

For a $p$-group $G$, and $H \le G$, if $G=H G^\prime$, then $H=G$ [on hold]

For a $p$-group $G$, and $H \le G$, prove that, if $G=H G^\prime$, then $H=G$; where $G^\prime$ is the commutator subgroup of $G$.
-1
votes
0answers
15 views

Subgroup $H \leq G$ acting on $G$ by translation is transitive?

In Elementary Topology. Textbook in Problems, by Viro, et al they state the following: Let $G$ be a topological group, $H \leq G$ a subgroup. Then $G$ is a homogeneous $H$-space under the ...
0
votes
0answers
20 views

Conjugacy classes of solvable groups [on hold]

If $A$ be a subset of solvable group $G$, let $k_G(A)$ be the number of $G$- conjugacy classes contained in $A$. Also let $N$ be a normal subgroup of $G$ of odd order, $\frac{G}{N}$ be an abelian ...
1
vote
2answers
47 views

Let G be an infinite cyclic group. Prove that G cannot have any non-identity elements of finite order.

SO I know that I'm suppose to prove it by contradiction and assume that the element has a positive power. I'm not really sure how to answer it though.
3
votes
0answers
35 views

Finite conjugate subgroup

In a paper titled "Trivial units in Group Rings" by Farkas, what does it mean by Finite conjugate subgroup. Here is the related image attached- What is finite conjugate subgroup of a group? It is ...
2
votes
2answers
36 views

If K and H are normal subgroups of $G$, $H \cap K = \{1\}$ and both $G/H$ and $G/K$ are abelian, then $G$ is abelian.

Let G be a group, and $H \trianglelefteq G$, $K \trianglelefteq G$. Prove that if $H \cap K = \{1\}$ and $ G / H $ and $ G/ K $ are abelian, then G is abelian. I've tried to give a proof by ...
182
votes
28answers
17k views

Nice examples of groups which are not obviously groups

I am searching for some groups, where it is not so obvious that they are groups. In the lectures script there are only examples like $\mathbb{Z}$ under addition and other things like that. I ...
0
votes
2answers
20 views

The number of $q$-Sylow subgroups cannot be $p$ for prime $p<q$

Since $q>p $, we cannot have $n_q=p $. Here $n_q $ is the number of $q $ Sylow subgroups. Why is the above statement true? This is a statement from Dummit and Foote.
0
votes
0answers
9 views

The relation of “be characteristic subgroup” is transitive? [duplicate]

If $K$ is a characteristic subgroup of $H$ and $H$ is a characteristic subgroup of $G$, then $K$, is characteristic in G? This is, the relation of "be characteristic" is transitive?
0
votes
0answers
12 views

p-primary component of a group

I have been asked to find the $3$-primary component of the group: $$\mathbb{Z_3}\oplus\mathbb{Z_5}\oplus\mathbb{Z_9}\oplus\mathbb{Z_{153}}$$ Now, I know that we define the $p$-primary component of a ...
1
vote
0answers
28 views

Identifying a group

When asked to identify the group $\mathbb{F}_4^{+}$, is my explanation below complete? If not, how can I complete it? The field $\mathbb{F}_4^{+} = \mathbb{F}_2[x]/(x^2+x+1)$ consists of the residues ...
2
votes
1answer
27 views

Show $\sigma^{-1} (i j)\sigma = ((i)\sigma (j)\sigma)$

Let $n \geq 2$ be an integer and $i, j \in \{1, 2, ..., n\} $ be distinct elements. Let $\sigma \in S_n$, Show that $\sigma^{-1} (i j)\sigma = ((i)\sigma (j)\sigma)$ let ...
2
votes
1answer
42 views

How to determine $\left|\operatorname{Aut}(\mathbb{Z}_2\times\mathbb{Z}_4)\right|$

I know that the group $\mathbb{Z}_2\times\mathbb{Z}_4$ has: 1 element of order 1 (AKA the identity) 3 elements of order 2 4 elements of order 4 I'm considering the set of all automorphisms on this ...
5
votes
2answers
53 views

If $|G|=p^n$, then $p^2 \le |G : G^\prime|$. [on hold]

Prove that, if $G$ be a p-group of order $p^n$, then $p^2 \le |G : G^\prime|$, where $G^\prime$ is the commutator subgroup of $G$ and $n \ge 2$.
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votes
1answer
35 views

Square element in a cyclic group

Which elements of a cyclic group are squares (an element $g$ of a group $G$ is a square if $g=h^2$ for some $h \in G$)? Here is my solution; is it correct? Let $G = \{ 1,a,a^2, \ldots , a^n \}$ ...
4
votes
1answer
43 views

Why is there only one group of order $n$ for some non-primes?

I would like to understand for which integers $n$ is there only one group of order $n$. (up to isomorphism). I understand that if $n$ is prime there is only one group of order $n$. In Sloane's OEIS ...
0
votes
1answer
17 views

Minimum possible size of generating set of $(\mathbb{Z}_p)^m$

Is it true that the group $(\mathbb{Z}_p)^m$ cannot be generated by less than $m$ elements?
3
votes
0answers
34 views

“Breaking the symmetry” in solving algebraic equations

I've heard somewhere a discussion about solving algebraic equations before: When solving a quadratic equation, we are essentially doing the following. Observe that ...
13
votes
1answer
371 views

Why does Group Theory not come in here?

Here is a list of questions that I find quite similar, for the one and only reason that they all show much "symmetry". Which is at the same time my problem, because I don't have a very precise notion ...
2
votes
2answers
40 views

Is that true for given group if their divisors are isomorphic, then their quotient group are isomorphic?

I think there will be counter examples in both cases: For given group $G$ and its normal group $H,K$ (1) If $H$ and $K$ are isomorphic then $G/H$ are $G/K$ are isomorphic (2) If $G/H$ are $G/K$ ...
-2
votes
1answer
47 views

Sylow's theorem and uniqunes of normal supgroup

Let $G$ be a finite group of order $pq,$ where $p$ and $q$ are primes such that $p < q.$ Then how to prove that $G$ has a unique normal subgroup of order $q?$
0
votes
2answers
566 views

Intersection of Normal Subgroups is Normal in Subgroup but Not Group - Fraleigh p. 143 14.35

Show that if H is a subgroup of a group G, and N is a normal subgroup in G, then $H \cap N$ is normal in H. Show by an example that $H \cap N$ need not be normal in G. I can condone the proof hence ...
1
vote
3answers
42 views

Cayley graph is a tree iff group is free

I am looking at this proof of this claim that the cayley graph is a tree iff g is a free group with generating set S. For the direction '$\implies $' I see that they have assumed that there are two ...
1
vote
0answers
29 views

Subgroups of finite index have finitely many conjugacy classes

is it true the following statement: Let $G$ be a group and let $H$ be a subgroup of $G$. If the index $[G:H]$ of $H$ in $G$ is finite, then $H$ have finitely many conjugacy classes. What I think, is ...
0
votes
2answers
56 views

Is it true that if $\sigma \in S_n,$ then $\sigma^n = \iota$?

I think I remember my abstract algebra professor mentioning in class that if $\sigma$ is any permutation belonging to the symmetric group $S_n,$ then $\sigma^n = \iota,$ the identity permutation. Is ...
-2
votes
0answers
35 views

how many proper subgroups are there in a trivial group{e}? [on hold]

Is there any proper subgroup of a trivial group of order 1? If yes or no how to prove it?
4
votes
1answer
59 views

When is $HK \cong H \times K$?

Suppose $G$ is a group and $H$ and $K$ are subgroups such that $G = HK$ and $H \cap K = \left\{e\right\}$, the identity element of $G$. When can we say that $HK \cong H\times K$? I tried to set up ...
2
votes
0answers
19 views

The relation of determinants between linear transformation.

I am studying the simplicity of PSL and came across the above statement. I don't understand why $\det L_c =c \det L$? (Given two set of basis of the same vectorspace, $v_1,...v_n$ and $w_1,...w_n$, ...
1
vote
1answer
61 views

Does $(\Bbb{Z}/n\Bbb{Z})^\times$ contain $0$?

I recently was told that $(\Bbb{Z}/n\Bbb{Z})^\times$ is cyclic if $n$ is prime. But then this is impossible if it contains $0$. Hence, does $(\Bbb{Z}/n\Bbb{Z})^\times$ contain $0$?
2
votes
0answers
34 views

Every normal subgroup of $GL_n(K)$ either contains $SL_n(K)$ or is contained in $Z$

In this page, it is claimed that if $K$ is a field then every normal subgroup of $GL_n(K)$ either contains $SL_n(K)$ or is contained in $Z$ (the center, which is the scalar matrices). What is the ...
4
votes
3answers
79 views

If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.

Let $G$ be a finite group. Suppose that $A$ and $B$ are to subsets of $G$. If $|G|<|A|+|B|$ prove that $$G=AB.$$
0
votes
0answers
17 views

Subgroups of direct product.

I have problem with finding all subgroups of $\mathbb{Z}_n \times D_m$ and $D_n \times D_m$. First, if $H_1 \leq \mathbb{Z}_n (\text{or} \ D_n)$ and $H_2 \leq D_m$, then $H_1 \times H_2 \leq ...
3
votes
1answer
50 views

Let $G$ be a finite group, $p$ the smallest prime divisor of $|G|$, and $x\in G$ an element of order $p$.

Suppose $h\in G$ is such that $hxh^{−1}=x^{10}$. Show that $p=3$. I am trying to solve this problem using group actions. Let $H$ and $X$ be the subgroups of $G$ generated by the elements $h$ and $x$, ...
2
votes
2answers
43 views

Is this group homomorphism well-defined?

Let $X = \langle a, b \mid aba^{-1}b^{-1} \rangle$ and $Y = \langle a, b \mid aba^{-1}b \rangle$. I want to define $f : X \rightarrow Y$, such that, $f(a) = a$ and $f(b) = b^2$, however I'm having ...
2
votes
0answers
25 views

Natural action of $\operatorname{Aut}(G)$ on sets of subgroups of $G$ of same order is transitive.

I am looking for the classification of those finite groups whose automorphism group acts transitively on sets of subgroups of same order. Let $G$ be a finite group and $d$ be a divisor of the order ...
3
votes
2answers
33 views

How to descent to smaller groups “by chopping off a node of the Dynkin diagram”?

I read in section 2 of this paper : "There is a well-defined chain to descent from $E_8$ to smaller groups by chopping off a node of the Dynkin diagram." What exactly is here referring to ...
3
votes
0answers
143 views
+50

A property of the subgroups lattices

Let $G$ be a finite group. Consider all the subgroups $H$ such that its subgroups lattice $\mathcal{L}(H)$ is distributive (i.e. the group $H$ is cyclic, by Ore's theorem), and among them, let $\{ ...
1
vote
2answers
45 views

Show that $Z(θ^G)≤H$

Suppose $H ≤ G$ and $θ \in Char(H)$. Show that $Z(θ^G)≤H$. ($Z$ is the centre and $θ^G$ is the character induced by $G$)
2
votes
1answer
44 views

If two subsets $S,T\subseteq G$ have sum of cardinalities greater than $|G|$, then $S+T=G$ [duplicate]

Let $S$ and $T$ are two subset of a finite group $(G,+)$ so that $|S|+|T|>|G|$, then Prove that $S+T=G$, where $S+T=\{s+t:s\in S ,t\in T\}$ My effort: It is clear that $S+T\subseteq G$ as ...
4
votes
2answers
100 views

rectangular groups are completely simple and orthodox

Let $S$ be a rectangular group. i.e $S$ is isomorphic to the direct product of a group and a rectangular band.
0
votes
0answers
25 views

Equivalence of right and left cosets of two different subgroups.

Let $A$ and $B$ be two (not necessarily equal) abelian subgroups of $S_5$. If $x$ is an element of $S_5$, under what condition is the following satisfied $$xA = Bx$$ Update: The original question I ...
0
votes
1answer
30 views

The Union of Two Normal Subgroups is also a normal subgroup

I know the statement The Union of Two Normal Subgroups is also a normal subgroup is false. Is there a counter example to show this? I can prove that the intersection is normal, but I can't disprove ...
2
votes
2answers
70 views

Proving that a set is infinite

Consider $G\subset M_2(\mathbb{C})$ where $G =\begin{Bmatrix} \begin{pmatrix} a & 10b\\ b & a \end{pmatrix} & | a,b \in \mathbb{Q},a^2-10b^2=1& \end{Bmatrix} $ . Prove that G is ...
3
votes
1answer
25 views

List the elements and cosets

In the group $\mathbb{Z}_{24}$, let $H=\langle 4 \rangle$ and $N=\langle6\rangle$ a. list the elements in $HN$ (usually write $H+N$ for these additive groups) and $H\cap N$ So I think $H=\langle4 ...
2
votes
4answers
242 views

Problem from Herstein (Group Theory)

This is the problem from Topics in Algebra by I. N. Herstein. Part of Example No. 2.2.9: Let $G$ be the set of all $2 \times 2$ matrices $ \left( {\begin{array}{cc} a & b \\ c & d \\ ...
3
votes
1answer
51 views

Second Isomorphism Theorem

There is one little detail in the proof I would very much like to get your opinion of. Look at where I have circled in red: There it seems that they have used that $\mu_2((hn)N)=h$. But isn't ...
0
votes
1answer
30 views

Normal subgroup corresponding to a relation

Suppose I have a free group on $n$ elements, $FX$, quotient-ed by an element (say, $\langle a, b \rangle/aba^{-1}b^{-1}$), how do I compute the normal subgroup $N$ of $FX$, such that $FX/N$ matches ...
0
votes
1answer
40 views

Finite abelian group - product of its elements [on hold]

Is there a finite abelian group G such that the product of the orders of all its elements is 2^2009?