In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is usually denoted H ≤ G, read as "H is a subgroup of G".
The trivial subgroup of any group is the subgroup {e} consisting of just the identity element.
A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G). This is usually represented notationally by H < G, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e}).If H is a subgroup of G, then G is sometimes called an overgroup of H.
The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group G is sometimes denoted by the ordered pair (G, ∗), usually to emphasize the operation ∗ when G carries multiple algebraic or other structures.
Hi together!
Say we have ## \Lambda_q{(A)} = \{\mathbf{x} \in \mathbb{Z}^m: \mathbf{x} = A^T\mathbf{s} \text{ mod }q \text{ for some } \mathbf{s} \in \mathbb{Z}^n_q\} ##.
How can we proof that this is a subgroup of ##\mathbb{Z}^m## ?
For a sufficient proof we need to check, closure...
So, a friend of mine has attempted a solution. Unfortunately, he's having numbers spawn out of nowhere and a lot of stuff is going on there which I can't make sense of. I'm going to write down the entire attempt.
$$
0 \in X \; \text{otherwise no subgroup since neutral element isn't included}...
Hi,
The textbook asks me to use subgroup axioms to prove why a set of permutations that interchange two specific symbols in S4 is or isn't a subgroup of the symmetric group, and the same for a set of permutations that fix two elements.
My guess is that the set of permutations that interchange...
Clearly e ∈ N. If a, b ∈ N, say ##a^k = b^l = e##, for some k,l ∈ N, then ##(ab)^{kl} = (a^k )^l (b^l )^k = e^l e^k = e##; thus, ab ∈ N. Also, ##|a|=|a^{−1}|##, so ##a^{−1}## ∈ N. Thus, N is a subgroup. As G is abelian, it is normal. Take any c ∈ G. If, for some n ∈ N, we have ##(cN)^n = eN##...
What is difference between subgroup and closed subgroup of the group? It is confusing to me because every group is closed.
In a book Lie groups, Lie algebras and representations by Brian C. Hall is written
"The condition that ##G## is closed subgroup, as opposed to merely a subgroup, should be...
Homework Statement
HiI am following this proof attached and am just stuck on the bit that says:
‘since ##\Omega## is a group it follows that ##|z-\omega|<2\epsilon ## contains..’Tbh, I have little knowledge on groups , it’s not a subject I have really studied in any of my classes-so the only...
Homework Statement
Find all cosets of the subgroup H in the group G given below. What is the index (G : H)?
H = <(3,2,1)>, G = S3
Homework EquationsThe Attempt at a Solution
I will leave out the initial (1,2,3) part of the permutation. We have S3 =...
Homework Statement
Let ##H## be a subgroup of ##G## and fix some element ##g\in G##.
Prove that ##gHg^{-1}=\{ghg^{-1} \mid h\in H\}## is a subgroup of ##G## of the same order as ##H##.
Deduce that if ##H## is the unique subgroup of ##G## of order ##|H|## then ##H\trianglelefteq G##.
Homework...
Let G be the group of symmetries (including flips) of the regular heptagon (7-gon).
As usual, we regard the elements of G as permutations of the set of vertex labels; thus, G ≤ S7.
(a) Let σ denote the rotation of the 7-gon that takes the vertex 1 to the vertex 2. Write down the cyclic...
Homework Statement
Let ##G## be a group and let ##A \subseteq G## be a set. The normal subgroup of ##G## generated by ##A##, denoted ##\langle A \rangle ^N##, is the set of all products of conjugates of elements of ##A## and inverses of elements of ##A##. In symbols,
$$\langle A \rangle ^N= \{...
In this PDF, http://www.math.unl.edu/~bharbourne1/M417Spr04/M417Exam2Solns.pdf,
in answering why a subgroup of index 2 is normal, the author says that the only two cosets must be ##A## and ##gA##. Why so? Why there can't be another element ##g'## such that ##G = g' A + g A##?
Homework Statement
I am looking at the quotient group G = Z/3Z which is additive and abelian. The equivalence classes are:
[0] = {...,0,3,6,...}
[1] = {...,1,4,7,...}
[2] = {...,2,5,8,...}
I want to prove [0] is a normal subgroup, N, by showing gng-1 = n' ∈ N for g ∈ G and n ∈ N. Since G...
Homework Statement
Prove that if ##f:G\to H## is a group homomorphism and ##K\leq H## then the preimage of ##K##, defined as ##f^{-1}(K)=\{g\in G | f(g)\in K\}##, is a subgroup of ##G##.
Homework EquationsThe Attempt at a Solution
1) Note that ##f^{-1}(K)## is nonempty, since ##f(e_G) = e_H##...
Homework Statement
Let ##R## be the set of all polynomials with integer coefficients in the independent variables ##x_1, x_2, x_3, x_4##.
##S_4## acts on ##R## by the group action ##\sigma \cdot p(x_1,x_2,x_3,x_4) = p(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)},x_{\sigma(4)})##. Exhibit all...
Problem: Let ##G=S_n##, fix ##i \in \{1,2, \dots, n \}## and let ##G_i = \{ \sigma \in G ~|~ \sigma (i) = i \}##. Use group actions to prove that ##G_i## is a subgroup of G. Find ##|G_i|##.
So here is what I did. Let ##A = \{1,2, \dots, n \}##. I claim that ##G## acts on ##A## by the group...
Homework Statement
Show that if ##H## is a subgroup of ##G##, then ##H \le N_G (H)##
Homework EquationsThe Attempt at a Solution
Essentially, we need to show that ##H \subseteq N_G (H)##; since they are both groups under the same binary operation the fact that they are subgroups will result...
Homework Statement
Let ##H_1 \le H_2 \le \cdots## be an ascending chain of subgroups of ##G##. Prove that ##H = \bigcup\limits_{i=1}^{\infty} H_{i}## is a subgroup of ##G##.
Homework EquationsThe Attempt at a Solution
Certainly ##H## is nonempty, since each subgroup ##H_i## has at least the...
Homework Statement
Let ##n \in \mathbb{Z}^+## and let ##F## be a field. Prove that the set ##H = \{(A_{ij}) \in GL_n (F) ~ | ~ A_{ij} = 0 ~ \forall i > j \}## is a subgroup of ##GL_n (F)##
Homework EquationsThe Attempt at a Solution
So clearly the set is nonempty since ##I_n## is upper...
Homework Statement
Let H and K be subgroups of G. Prove that if ##H \cup K## is a subgroup of ##G## then ##H \subseteq K## or ##K \subseteq H##
Homework EquationsThe Attempt at a Solution
Suppose that ##H \cup K \le G##. For contradiction, suppose that neither H nor K is a subset of the...
Homework Statement
A nonzero free abelian group has a subgroup of index ##n## for every positive integer ##n##
Homework EquationsThe Attempt at a Solution
If ##F## is a nonzero free abelian group, then ##F## is isomorphic to the direct sum ##G= \sum_{i \in I} \Bbb{Z}##, where ##I \neq...
I was reading a Wikipedia page where it's given an example of a group that's not a Lie Group. Here's the page https://en.wikipedia.org/wiki/Lie_group ; refer to "Counterexample".
If we work with the topology of ##\mathbb{T}^2## it seems obvious that a map from some ##\mathbb{R}^m## would not be...
Homework Statement
Let G be a group. Let H and K be subgroups of G. Prove that if
H ##\subseteq## K, then H is a subgroup of K.
Homework EquationsThe Attempt at a Solution
H is a subset of K and H,K are groups.
if x,y, xy ##\epsilon## H, then x,y, xy ##\epsilon## K.
So H is closed under...
Let H be a subgroup of G and let L be a subgroup of H. Prove that K is a subgroup of G.
This question seems very redundant to me, isn't anything in a subgroup automatically a subgroup of anything the larger group is a subgroup of. Can some one explain this proof to me?
Let G be a group and let H be a subgroup.
Define N(H)={x∈G|xhx-1 ∈H for all h∈H}. Show that N(H) is a subgroup of G which contains H.
To be a subgroup I know N(H) must close over the operations and the inverse, but I am not sure hot to show that in this case.
Homework Statement
D4 acts on the vertices of the square. Labeling them counterclockwise
starting from the top left as 1, 2, 3, 4, find the corresponding homomorphism
to S4.
Homework EquationsThe Attempt at a Solution
I am not completely sure what the question is asking. It's pretty clear to...
Let ##G## be a group. I have shown that ##H_a = \{x \in G | xa=ax \}## is a subgroup of G, where ##a## is one fixed element of ##G##. I am now asked to show that ##H_S = \{x \in G ~| ~xs=sx,~ \forall s \in S\}## is a subgroup of ##G##. How would proving the former differ from proving the latter...
Hello,
Just a really quick question on definition of discrete subgroup.
This is for an elliptic functions course, I have not done any courses on topology nor is it needed, and most of the stuff I can see online refer to topology alot, so I thought I'd ask here.
I need it in the complex plane...
Homework Statement
Prove that Q+, the group of positive rational numbers under multiplication, is isomorphic to a proper subgroup of itself.
Homework Equations
None
The Attempt at a Solution
[/B]
Not at all sure if this is legit.
Let phi: Q+ --> G
phi(x) = x2, x is in Q+
We will...
Hey! :o
Let $\rho=\sqrt[3]{\frac{1+\sqrt{5}}{2}}$.
We have that $\rho$ is a root of $f(x)=x^6-x^3-1\in \mathbb{Q}[x]$, that is irreducible over $\mathbb{Q}$.
We have that all the roots of $f(x)$ are $\rho, \omega\rho, \omega^2\rho, -\frac{1}{\rho}, -\frac{\omega}{\rho}...
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Let $E/F$ be a finite Galois extension and let the chain of extensions $F =
K_0 \leq K_1 \leq \dots \leq K_n = E$.
Let $G = Gal(E/F)$ and, for $i = 0, 1, \dots , n$, let $H_i$ be the subgroup of $G$, that corresponds to $K_i$ through the Galois mapping.
I want to show that, for any...
Hello! I have this problem:
If H is a subgroup of prime index in a finite group G, show that either H is a normal subgroup or N(H) = H.
What does N(H) means? I don't want a solution for the problem (at least not yet), I just want to know what that notation means. Thank you!
Homework Statement
Let G be a group. An Isomorphism Φ: G --> G is called an automorphism of G. Let Aut(G) denote the set of all automorphisms of G.
Prove that Aut(G) is a subgroup of Bij(G).
Homework Equations
For it to be a subgroup we need to show:
i) e ∈ Aut(G)
ii) For all x,y ∈ Aut(G)...
Hey! :o
Let $P$ be a $p$-Sylow subgroup in $G$ and $N=N_G(P)$.
I want to show that there is no proper normal subgroup $H$ of $G$ that contains $N$.
We suppose that there is a proper normal subgroup $H$ of $G$ that contains $N$, $$N\leq H<G$$
Then $[G:N]=[G:H][H:N]$, with $[G:H]>1$.
How...
Hey! :o
Let $G$ be a non-abelian finite group with center $Z>1$.
I want to show that if $G/Z$ is a $p$-group, for some prime $p$, then $G$ contains a normal $p$-Sylow subgroup and $p\mid |Z|$.
We have that $$|G/Z|=p^n, n\geq 1\Rightarrow \frac{|G|}{|Z|}=p^n\Rightarrow |G|=p^n|Z|$$ That means...
Hi. I have the following question:
Let G be a finite group. Let K be a subgroup of G and let N be a normal subgroup of G. Let P be a Sylow p-subgroup of K. Is PN/N is a Sylow p-subgroup of KN/N?
Here is what I think.
Since PN/N \cong P/(P \cap N), then PN/N is a p-subgroup of KN/N.
Now...
Hey! :o
I want to show that if $S\in \text{Syl}_p(G)$ and $N\trianglelefteq G$, then $N\cap S\in \text{Syl}_p(N)$. Could you give me some hints how we could show that? (Wondering)
Do we maybe use Frattini's Argument? (Wondering)
From that we have that since $N\trianglelefteq G$ and $S\in...
How do I find the left cosets of:
$(a)$ $\left\{ [0], [5], [10] \right\} \le \mathbb{Z}_{15}$ ($\mathbb{Z}_n$ is additive group modulo $n$).
$(b)$ $\left\{e, y, y^2, y^3 \right\} \le D_4$ where $y$ denotes rotation of a square.
The not equal to here denotes subgroup. The trouble I've with...
Hello!
As far as I know any subgroup can, in principle, be used to divide group into bundle of cosets. Then any group element belongs to one of the cosets (or to the subgroup itself). And such division still is not qualified as a quotient.
Yes, the bundle of cosets in this case will be...
Let $S$ be any set, $A(S)$ the set of one-to-one mappings of $S$ onto itself, made into a group under the composition of mappings. If $x_0 \in S$, what is meant by $H(x_0) = \left\{\phi \in A(S): x_0 \phi = x_0\right\}$? The set that contains the element $\phi$ in $A(S)$ that maps $x_0$ onto...
Hey! :o
I want to show that $N\{1, (12)(34), (13)(24), (14)(23)\}$ is a normal subgroup of $S_4$ that is contained in $A_4$ and that satisfies $S_4/N\cong S_3$ and $A_4/N\cong Z_3$. Let $\sigma\in S_4$.
We have the following:
$$\sigma 1 \sigma^{-1}=\sigma (1) \\ \sigma (1 2)(3 4)...
Hey! :o
I want to show that if $H$ is a cylic normal subgroup of a group $G$, then each subgroup of $H$ is a normal subgroup of $G$.
I have done the following:
Since $H$ is a normal subgroup of $G$, we have that $$ghg^{-1}=h\in H, \ \forall g \in G \text{ and } \forall h\in H \tag 1$$...
Hello,
Suppose we have a semigroup S with a subgroup G≤S.
Assume there is an element s∈S that commutes with all the elements in G. Does this statement implies (or is equivalent to) another statement?
If hypothetically the element s would have been in G, then we could have said that s was an...
Homework Statement
Homework Equations
subgroup axioms:
1. a, b in T(G), then ab in T(G)
2. existence of identity element.
3. a in T(G), then a^-1 in T(G)
The Attempt at a Solution
1.
let a be in T(G), then a^n = e.
let b be in T(G), then b^n = e
(ab)^n = (a^n)(b^n) = (e)(e) = e
axiom 1...
Homework Statement
Let ##G## be a group and ##\sim## and equivalence relation on ##G##. Prove that if ##\sim## respects multiplication, then ##\sim## is the equivalence relation associated to some normal subgroup ##N\trianglelefteq G##; i.e., prove there is a normal subgroup ##N## such that...
Note: I only need help on the underlined portion of the problem, but I'm including all parts since they may provide relevant information. Thanks in advance.
1. Homework Statement
Let S be a subset of a group G such that g−1Sg ⊂ S for any g∈G. Show that the subgroup ⟨S⟩ generated by S is...
Homework Statement
Find a normal subgroup H of Zmn of order m where m and n are positive integers. Show that H is isomorphic to Zm.
Homework EquationsThe Attempt at a Solution
I am honestly not even sure where to start. My initial thoughts were if Zmn was isomorphic to Zm x Zn then I could...
Let H be a subgroup of G, then:
Core H = {a in G | a is an element of gHg^(-1) for all g in G} = The intersection of all conjugates of H in G
My book goes on to say that every element of Core H is in H itself because H is a conjugate to itself. Previously, I understood that H was a conjugate to...
I am self-studying a class note on finite group and come across a problem like this:
PROBLEM: Let ##G## be a dihedral group of order 30. Determine ##O_2(G),O_3(G),O_5(G), E(G),F(G)## and ##R(G).##
Where ##O_p(G)## is the subgroup generated by all subnormal p-subgroups of ##G##; ##E(G)## is the...