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

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Are there problems at the interface of group theory and calculus?

I wonder whether there are mathematical problems that require the joint use of group theory and calculus? Can someone please give me an example if there are any?
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42 views

Are these equivalent definitions of the Grothendieck group construction?

I'm learning K-theory and I'm slightly confused. Given an abelian monoid $M$, we can construct the Grothendieck group $$ G(M) = (M \times M)/\sim$$ and I've seen two definitions for the equivalence ...
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89 views

Counterexample to a weak “definition” of group

Usually, we see a group as the structure $\langle G, \cdot\rangle$, where $G\neq\emptyset$ and the binary operation $\cdot$ is associative and $\exists e\in G,\,\forall a\in G,\,(a\cdot e=a=e\cdot ...
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2answers
110 views

How to prove $\frac{1}{q} \binom{p^k q}{p^k} \equiv 1 \mod p\;\;$?

An argument that I just came across asserts that if $p$ is prime, and $k, q\in \mathbb{Z}_{>0}$, then $$\frac{1}{q} \binom{p^k q}{p^k} \equiv 1 \mod p \;.$$ This assertion was made in a way (i.e. ...
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1answer
48 views

Show that $P \cong C_p \wr C_p$

Let P be a Sylow-p-subgroup of $S_{p^2}$. I need to show that $P \cong C_p \wr C_p$. I didn't get very far yet. I have already proven that $|P|=|C_p \wr C_p|=p^{p+1}$.
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35 views

Infinite dimension spaces

Consider, doubly infinite row vectors $(a)=(\cdots, a_{-1}, a_0, a_1,\cdots)$ with $a_i's$ real form a vector space. Is this space isomorphic to $\mathbb{R^\infty}$. Here ...
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1answer
145 views

Subset of all invertible elements is a group

Let $S$ be a set with an associative law of composition and with an identity element. Show that the subset $S^*$ $\subset$ $S$ consisting of all invertible elements is a group. Attempt at proving: We ...
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1answer
54 views

cyclic group - automorphism

If $G$ is a cyclic group where both $a$ and $b$ are generators. How would I prove that $f:G \rightarrow G$ given by $f(a^i)=b^i$ is an automorphism. I know that an automorphism is the identity map. ...
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1answer
50 views

Directly Computable Homomorphism

We call a homomorphism $f$ defined on a permutation or a matrix group $G$ directly computable if there is an efficient method of evaluating $f(g)$ directly from $g$, for all $g$ in $G$. I can not ...
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2answers
143 views

If $a,b \in G$ and $ab=ba$, prove $ (ab)^{|a||b|}=e$.

I need help with "Show that the above may be false if $ab\neq ba$". Thanks for the help!!
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2answers
44 views

Centralizer as a subgroup.

Let $G$ be a group and $a\in G$. Define the centralizer of $a$ to be $\hspace{150pt} C(a)=\{g\in G : ga=ag\}$. That is, $C(a)$ consists of all the elements that commute with $a$. Show that $C(a)$ ...
2
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0answers
61 views

Groups of order 8

Ok so I am looking at a proof to all the groups of order 8, I've attached an image which basically is the start of the proof. In particular the part highlighted with yellow is causing me problem, I ...
2
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2answers
138 views

Show that $S_3$ is presented by $\langle a,b\mid a^3, b^2,ab=ba^2\rangle$

Show that group $S_3$ of the objects $x,y,z$ is presented by $\langle a,b \mid a^3,b^2,ab=ba^2\rangle$ under the mapping $a \to (xyz)$ , $ b\to (xz)$ I'm confused to what is to be shown in these ...
1
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1answer
55 views

existence of $a \neq e$ in group such that $a^2=e$ [duplicate]

$G$ - group, $|G|=2k$, $k\in \Bbb N$ Does there exist $a\in G; a \neq e: a^2 =e$? I think I should somehow use the fact, that there is odd number of elements of $G$ which are not $e$.
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0answers
43 views

Counting technique in the group theory

I want to prove : If $H$ and $K$ are subgroups of a group $G$, then $\left [ H:H \cap K \right ]\leq[G:K]$. My proof is : $\left[ H:H\cap K \right]=\frac{\left | H \right |}{\left| H\cap K ...
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2answers
90 views

if $A^\times $ is a commutative group, does $A$ necessarily be a commutative ring?

Let $A$ be a ring and $A^\times$ be the collection of unit elements of $A$. If $A$ is a commutative ring, then $A^\times$ is a commutative group. Conversely, if $A^\times $ is a commutative group, ...
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2answers
100 views

Fundamental group of Poincaré sphere

Do the two presentations below, $$G=\langle d,v \mid dv^2d=vdv, dv^3d=v^2 \rangle$$ and $$\langle r,s,t \mid r^2=s^3=t^5=rst \rangle = \langle s,t \mid (st)^2=s^3=t^5 \rangle,$$ define the same group? ...
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1answer
32 views

Finding the inverse of P on the generalized weierstrass equation

If P = (x, y) = ∞ is on a monic cubic polynomial, then −P is the other finite point of intersection of the curve and the vertical line through P. Show that −P = (x, −$a_{1}x$ − $a_3$ − y). (Hint: This ...
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0answers
100 views

Why is the special linear group generated by elementary matrices that add a multiple of row$ j$ to row $i$

The general linear group is generated by elementary matrices that add a multiple of row $j$ to row $i$ and elementary matrices that multiply row $i$ by a scalar. This is because you can write an ...
2
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1answer
57 views

If $H\triangleleft G$ then $Z(H)\leq Z(G)$.

If $H\triangleleft G$ then $Z(H)\leq Z(G)$. Is this true?I need so help for starters.
2
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0answers
52 views

Terminology problem

I am not a mathematician, please forgive my incorrect language. My question involves terminology. If a finite non-abelian group G is represented by a set of unitary operators ${\mathbf G}_r, r = ...
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1answer
112 views

Find period of power sequence $a^k \mod m$, with $a, m$ not coprime

Let $a, m$ be positive integers and $m > 1$. I'm interested in the sequence $(a^k)_{k \in \mathbb{N_0}} \mod m$. Since there are only $m$ different values that can occur in the sequence and since ...
0
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1answer
64 views

Subgroups of index 3 are normal [duplicate]

$G$ has no subgroup with index 2. How can I show that all subgroups with index 3 are normal? (The index of the subgroup is the order of the quotient set.)
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0answers
36 views

center of the group of orthogonal matrices of dim 3 [duplicate]

i am looking for the center of the group of orthogonal matrices of dimension 3. i'm thinking it contains all rotations and reflections but i'm not sure i'm correct and (assuming i am) don't know how ...
3
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1answer
62 views

Condition to be a group.

Let $G$ be a semigroup. I'm showing that $G-group \iff [ \ \exists_{e\in G} \forall_{a\in G}: ea=a\ ] $ and $ [\ \forall_{a\in G}\exists_{a^{-1}\in G}: a^{-1}a=e \ ] $ "$\Rightarrow$" is obvious. ...
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1answer
78 views

group of automorphisms of vector spaces

let α is an automorphism of G as a vector space and β is an automorphism of H as a vector space as well, then the tensor product of G and H is an automorphism of G tensor H? Let suppose that G and H ...
2
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4answers
56 views

$H_1\times H_2 <G_1\times G_2 \Rightarrow H_1<G_1, H_2<G_2$?

Let $H_1, H_2, G_1, G_2$ be groups. Clearly if $H_1<G_1$ and $H_2<G_2$, then $H_1\times H_2<G_1\times G_2$. I'm wondering if the converse statement is true. I'm quite sure it's not. Can you ...
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1answer
70 views

Irreducibility of complex 2-dimensional character of the group $ S_3 $

Let $\chi$ be 2-dimensional complex character of the group $ S_3 $. Prove that $\chi$ is irreducible character iff $\chi((123))=-1$. There is hint in my book: " Use the Maschke's theorem and ...
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1answer
43 views

find a special element of $Sp(2n,q)$

Please help me about this question: I want to show that symplectic group $Sp(2n,q)$ has an element of order $q^n-1$.
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1answer
24 views

Find an abelian subgroup of $GL(2n, q)$ of special order

I want to prove that $GL(2n, q)$ where $q$ is even, has an abelian subgroup of order $q^{2n - 1}$. please help me.
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1answer
63 views

Does every group have a representation?

For any set $A$, we can give it an group structure and make it a free group. For example: $$\mathbb Z=<a;aa^{-1}=1>$$ Further more, we can introduce some relation on it: $$\mathbb ...
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1answer
41 views

Cyclic Subgroup?

For $U(16) = \{1,3,5,7,9,11,13,15\},$ is there a simple way to find $m \in U(16)$ such that $|m| = 4$ and $|\langle m\rangle \cap \langle 3\rangle| = 2$ and $m$ is unique without listing everything ...
4
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1answer
56 views

Submonoid: Identity

There's a common mistake appearing in quite many books in definitions and proofs as well; now my idea is maybe you can safe the day: Imagine you're having a monoid and a subset being closed under the ...
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1answer
23 views

Identifying some cyclic subgroup

Is there a fast way to argue that (for $a,b>1$ integers) the set of all $x\in\mathbf{Z}/b\mathbf{Z}$ with $ax=0$ is isomorphic to $\mathbf{Z}/{gcd(a,b)}\mathbf{Z}$? Maybe by counting the elements, ...
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0answers
51 views

Group structure on finite set.

A set $S$ consisting of $k^{k-1}$ elements is given. $k$ is a natural number greater than $1$. Can we give a group structure on it for every $k$? How many groups are there of order $k^{k-1}$ up to ...
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1answer
74 views

Is this group: $\mathbb Z^{*}_{15}$, is cyclic? [closed]

Is this group: $\mathbb Z^{*}_{15}$, is cyclic? I tried to find a generator. and i didn't found one. But how can i prove that this group is not cyclic?
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1answer
107 views

Injection function and product of two exponential elements - homomorphisms -

[Fraleigh, p.133, ex. 13.7] Let $f_i: G_i \rightarrow G_1 \times G_2 \times \dots \times G_r$ be given by $f_i(g_i) = (e_1, e_2, ..., g_i, ..., e_r),$ where $g_i \in G_i$ and $e_j$ is the ...
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1answer
61 views

Image of Group Homomorphism is Finite and Divides |Domain of Group| - Fraleigh p. 135 13.44

Let $\phi: G \rightarrow G'$ be a homomorphism. Show that if $|G|$ is finite, then $|\phi[G]|$ is finite and divides $|G|$. Because $φ[G] = \{φ(g) \, | \, g ∈ G\}$, we see $|φ[G]| ≤ \quad |G|$ which ...
1
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1answer
98 views

number of p-sylow subgroups(NBHM-$2014$)

Given a finite group and a prime $p$ which divides its order, Let $N(p)$ denote the number of $p-$sylow subgroups of $G$. If $G$ is a group of order $21$, what are the possible values of $N(3)$ and ...
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4answers
169 views

Prove that a cyclic group can have no more than one element of order two.

(1)Why can't a cyclic group have more than one element of order two? (2)Why does the group $U(n^2 -1)$ have to have more than one element of order two?
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1answer
68 views

If $H$ is a subgroup of $G$ ,why does $|Ha|=|H|=|aH|$ for all $a\in G$

Proof in the book notes:$Ha \rightarrow H$ is a bijection. END I think there may exist $x,y \in H$ such that $xa=ya= \alpha \in H$,then $xa \rightarrow \alpha ,ya \rightarrow \alpha$ it become a ...
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1answer
43 views

subgroups of ${Z}_{m}\oplus {Z}_{n}$

can someone please help me with solving this one ? let $m,n$ be integers with $gcd(m,n)=1$ and $G$ be the group $G = {Z}_{m}\oplus {Z}_{n}$ prove that any subgroup $H\leq G$ equals to: ...
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1answer
96 views

Visualize left, right cosets and conjugation

I drew everything that's in orange. Figure 6.8. Left illustration - Each left coset gH is where H arrows can reach from g, which looks like a copy of H based at g, as in the left illustration. ...
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1answer
41 views

Visualize cosets of $\left<(0,1)\right>$ partition $C_3 \times C_3$

Page 105 says - A careful look at Figure 6.9 reveals that the cosets of $\left< \, (0,1) \,\right>$ partition $C_3 \times C_3$. How is this true? The picture shows $gH = left picture = ...
4
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1answer
47 views

Visualize $C_2 \times C_4$ is normal subgroup

Page 120 says: Given our recent work with subgroups, you may have noticed that $C_2$ is a subgroup of $C_2 \times C_4$; specifically, it is the subgroup $<(1,0)>$. Furthermore, the cosets of ...
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1answer
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Visualize $A_4$ and $\langle x, z\rangle$ isomorphic to the Klein 4 group

Page 136 says Following Step 1 of Definition 7.5, the top of Figure 7.23 shows $A_4$ organized by the subgroup $\langle x, z\rangle$ (which is isomorphic to the Klein $4$ group. This reorganization ...
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68 views

Sylow subgroups

Let $G$ be a groups of order $385$ 1. Show that $P_7$,$P_{11}\triangleleft G$ 2. Show that $P_7 \subseteq Z(G)$ 3. Show that $Z(G)=P_7$ or that $G$ is cyclic *for $P_i$ - the sylow-$i$ subgroups of ...
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2answers
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Identifying a quotient group (NBHM-$2014$)

Let $\mathbb C^*$ denote the multiplicative group of non-zero complex numbers and let $P$ denote the subgroup of positive real numbers. Identify the quotient group. My thought $$\frac{\mathbb ...
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1answer
64 views

Does there always exists an injective homomorphism from $G$ into $S_m$ for $m<n$ [duplicate]

Does there always exists an injective homomorphism from $G$ into $S_m$ for some $m<n$ where $|G|=n$ I know that for $|G|=n$ there is always an injective homomorphism from $G$ into $S_n$
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2answers
80 views

Proving two groups are isomorphic.

Let $G$ be an abelian group, and let $f:G\rightarrow \mathbb{Z}$ be an epimorphism (surjective homomorphism). Prove that there exist a subgroup $H\subseteq G$ such that $H\cong \mathbb{Z}$, and ...