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.
G,H be groups(finite or infinite)
Prove that if (G:H)=n, then there exist some normal subgroup K of G (G:K)≤n!
example) let G=A5, H=A4 then (G:H)=5, then K={id} exists, (G:K)≤5!
Homework Statement
i just don't really get the one step subgroup test, which is very important, and something i should understand. can someone walk me through in general how to use the test? maybe give me a simple example? thanks.
Homework Equations
The Attempt at a Solution
I am somewhat distracted so this post will not be what it should, given that GAP is one of my interests.
For those who don't already know: GAP is a powerful open source software package for computational algebra, especially computational group theory and allied subjects. This long running...
Hi...
:smile:
I need a program that can give me all the subgroups of a group that I define. I also need it to give me the names of the subgroups as per some predefined library.
I tried GAP. It gives me the subgroups, but each subgroup is represented by a list of generators. There...
Homework Statement
Let R = {all real numbers}. Then <R,+> is a group. (+ is regular addition)
Let H = {a|a \epsilon R and a2 is rational}.
Is H closed with respect to the operation?
Is H closed with respect to the inverse?
Is H a subgroup of G?
Homework Equations
N/A
The Attempt at a...
Homework Statement
Is the Cyclic Subgroup { (1), (123), (132)} normal in A_{4} (alternating group of 4)
Homework Equations
The Attempt at a Solution
So I believe if I just check if gH=Hg for all g in A_4 that would be suffice to show that it is a normal subgroup, but that seems...
Homework Statement
If H is a subgroup of G, show that g^{-1}Hg={g^{-1}hg \; h\in H is a subgroup for each g\in G
Homework Equations
The Attempt at a Solution
I know I just have to check for closure and inverses, but the elements in this group g^{-1}hg with different h or with...
Where n >= 2.
Is this true or false? I only got so far:
If K is a subgroup of index 2, then it's normal. K is normal in Sn, so it's a union of conjugacy classes. Also, since |An K| = |An| |K| / |An intersection K| = 1/2n! * 1/2n! / |An intersection K| <= n!, then 1/4n! <= |An intersection...
It is said on wiki* that
"Maximal compact subgroups are not unique unless the group G is a semidirect product of a compact group and a contractible group, but they are unique up to conjugation, meaning that given two maximal compact subgroups K and L, there is an element g in G such that...
Homework Statement
List the elements of the subgroup generated by the given subset:
The subsets {4,6} of Z12
Homework Equations
The Attempt at a Solution
so the GCD of 4 and 12 is 4, 12/4 = 3 elements that <4> generates : {0, 4, 8}
GCD of 6 and 12 is 6, 12/6 = 2 elements...
Homework Statement
If G contains a normal subgroup H which is isomorphic to \mathbb{Z}_2, and if the corresponding quotient group is infinite cyclic, prove that G is isomorphic to \mathbb{Z}\times\mathbb{Z}_2
The Attempt at a Solution
G/H is infinite cyclic, this means that any g\{h1,h2\} is...
Hi
I'd like to show that SU(n) is a normal subgroup of U(n).
Here are my thoughts:
1)The kernel of of homomorphism is a normal subgroup.
2)So if we consider a mapping F: G-> G'=det(G)
3)Then all elements of G which are SU(n), map to the the identity of G', therefore SU(n) is a...
Homework Statement
Let H be a finite subgroup of a group G. Verify that the formula (h,h')(x)=hxh'-1 defines an action of H x H on G. Prove that H is a normal subgroup of G if and only if every orbit of this action contains precisely |H| points.
The Attempt at a Solution
I solved the first...
Homework Statement
If G is a finite group which acts transitively on X, and if H is a normal subgroup of G, show that the orbits of the induced action of H on X all have the same size.
The Attempt at a Solution
By the Orbit-Stabilizer theorem the size of the orbit induced by H on X is a...
Homework Statement
Let G be a group and let a, b be two fixed elements which commute with each other (ab = ba). Let H = {x in G | axb = bxa}. Prove that H is a subgroup of G.
Homework Equations
None
The Attempt at a Solution
I'm using the subgroup test. I know how to show...
Ok, this is a really easy question, so I apologise in advance.
Let A be an abelian subgroup of a topological group. I want to show that cl(A) is also.
Now I've shown that cl(A) is a subgroup, that is fairly easy. So I just need to show it is abelian.
For a metric space, it is easy...
'Prove that if a finite group G has only one maximal subgroup M, then |G| is the power of a prime'
I've somehow deduced that no finite group has only one maximal subgroup, and I'm having trouble seeing where I went wrong.
This is what I have:
Let H_1 be a subgroup of G. Either H_1 is...
How would one go about proving a particular subset of S4 is a normal subgroup of S4? Since S4 has 24 elements, I'm wondering if there is any other way to prove this other than a brute force method.
Homework Statement
S4xZ/5
Find the number of p-Sylow subgroups for p=2, 3, and 5
The Attempt at a Solution
S4 is no problem. There I can use some binomial coefficients and count stuff. I was wondering how to tackle any Z/n.
2-Sylow
First we find 2-sylow subgroups of S4...
I need to find just one non-abelian subgroup of size 6 in A_6.
I have started by noting that the subgroup must be isomorphic to D_6 and then tried to use the permutations in D_6 that sends corners to corners. I then came across the problem that the reflection elements in D_6 consist of...
Homework Statement
H and K are subgroups of G. Prove that H\capK is also a subgroup.
The Attempt at a Solution
For H and K to be subgroups, they both must contain G's identity. Therefore, e \in H\capK. Therefore, H\capK is, at least, a trivial subgroup of G.
This was a test...
Homework Statement
G is a finite p-group, show that G/ \Phi (G) is elementary abelian p-group.
Homework Equations
\Phi (G) is the intersection of all maximal subgroups of G.
The Attempt at a Solution
By sylow's theorem's we have 1 Sylow p-subgroup which is normal, call P. Then the order...
Given a subgroup of G=S3={(1)(2)(3), (1 2)(3)} acting on the set S3 defined as g in G such that gxg-1 for every x in S3. Describe the orbit.
The first one is (1)(2)(3)x(3)(2)(1). This orbit is just the identity.
For the second one, I'm not sure how to describe (1 2)(3) except by...
This specifically relates more towards the argument as to why an inverse exists.
First the problem
The normalizer is defined as follows, NG(H)={g-1Hg=H} for some g in NG(H). I get why identity exists and why the operation is closed. It is in arguing that an inverse exists that I have beef...
The problem is to verify that {(1), (1 2), (3 4), (1 2)(3 4)} is an Abelian, noncyclic subgroup of S4.
I was able to show that it is Abelian through pairing the permutations, but my mind stopped at the noncyclic part. When showing that a group is cyclic or noncyclic, what exactly do I have to...
Theorem:
A subset H of a group G is a subgroup of G if and only if:
1. H is closed under binary operation of G,
2. The identity element e of G is in H,
3. For all a \in H it is true that a^{-1} \in H also.
Proof:
The fact that if H \leq G then Conditions 1, 2, and 3 must hold follows at one...
Hello,
given a (semi)group A and a sub-(semi)group S\leq A, I want to define a morphism f:A\rightarrow A such that f(s)\in S, for every s \in S.
Essentially it is an ordinary morphism, but for the elements in S it has to behave as an endomorphism.
Is this a known concept? does it have already a...
Homework Statement
This is an exercise from Jacobson Algebra I, which has me stumped.
Let G = G1 x G2 be a group, where G1 and G2 are simple groups.
Prove that every proper normal subgroup K of G is isomorphic to G1 or G2.
Homework Equations
The Attempt at a Solution
Certainly...
Homework Statement
If J is a subgroup of G whose order is a power of a prime p, verify that J must be contained in a Sylow p-subgroup of G. The problem says to refer to a lemma that given an action by a subgroup H on its own left cosets, h(xH)=hxH, H is a normal subgroup iff every orbit of the...
Homework Statement
S.T <f> is not normal. where f is a reflection
Homework Equations
<f>={e,r^0 f, r^1f,r^2f,..}
WTS For any g in D-n, g(r^kf)g^-1 Not In <F>
The Attempt at a Solution
Elements of D-n are r^k, r^kf
For r^k, (r^k)(r^if)(r^-k) is in <f>.
So I am stuck
Homework Statement
Let Dn = {1,a,..an-1, b, ba,...ban-1} with |a|=n, |b|=2,
and aba = b.
show that every subgroup K of <a> is normal in Dn.
The Attempt at a Solution
First, we show <a> is normal in Dn.
<a> = {1,a,...an-1} has index 2 in Dn and so is normal
by Thm (If H is a subgroup...
The integers Z are a normal subgroup of (R,+). The quotient R/Z is a familiar topological group; what is it?
okay... i attempted this problem...
and I don't know if i did it right... but can you guys check it?
Thanks~
R/Z is a familiar topological group
and Z are a normal subgroup of...
Homework Statement
A. Let |g| = 20 in a group G. Compute
|g^2|, |g^8|,|g^5|, |g^3|
B. In each case find the subgroup H = <x,y> of G.
a) G = <a> is cyclic, x = a^m, y = a^k, gcd(m,k)=d
b) G=S_3, x=(1 2), y=(2 3)
c) G = <a> * <b>, |a| = 4, |b| = 6, x = (a^2, b), y = (a,b^3)
The...
Homework Statement
If G is an abelian group, show that H = { a in G | a^2=1} is a subgroup of G.
Give an example where H is not a subgroup.
The Attempt at a Solution
For showing H is a subgroup of G, hh' in G and h^-1 in G.
(a^2)(a^2) in G also a = a^-1 in G so H is a subgroup of G...
This is probably very trivial, but I can't find an argument, why the orthochronal transformations (i.e. those for which \Lambda^0{}_0 \geq 1) form a subgroup of the Lorentz group, i.e. why the product of two orthochronal transformations is again orthochronal?
Since when you multiply two...
The parts of this problem form a proof of the fact that if G is a finite subgroup of F^*, where F is a field (even if F is infinite), then G cyclic. Assume |G|=n.
(a) If d divides n, show x^d-1 divides x^n-1 in F[x], and explain why x^d-1 has d distinct roots in G.
(b) For any k let \psi(k)...
Let p be a prime, G a finite group, and P a p-Sylow subgroup of G. Let M be any subgroup of G which contains N_G(P). Prove that [G:M]\equiv 1 (mod p). (Hint: look carefully at Sylow's Theorems.)
Homework Statement
Assume that the nonzero elements of Z13 form a group G under multiplication [a][b] = [ab].
a) List the elements of the subgroup <[4]> of G, and state its order
The Attempt at a Solution
So I thought this would be like some of the previous problems.
I assumed...
Homework Statement
If H ≤ G is cyclic and normal in G, prove that every subgroup of H is also normal in G.
The attempt at a solution
Let H = <h>. We know that for g in G, hi = ghjg-1 by the normality of H. A simple induction shows that hin = ghjng-1, so that <hi> = g<hj>g-1. Now all I need...
Hi, I was reading Cartan's Theorem:
A Group H is a Lie Subgroup to Lie Group G if H is a closed subgroup to G.
Now first of all, is this a definition of Lie Subgroup?
Second, what does it mean that the subgroup is "closed"? I thought all groups where closed under group multiplication...
Homework Statement
Let H, K be subgroups of S_5, where H is generated by (1 2 3) and K is generated by (1 2 3 4 5). Is HK a subgroup of S_5?
Homework Equations
HK is a subgroup iff HK = KH.
The Attempt at a Solution
Is there an easy way of answering this question without computing HK...
Homework Statement
If G is a group of order 231, prove that the 11-Sylow subgroup is in the center of G.
The attempt at a solution
The number of 11-Sylow subgroups is 1 + 11k and this number must be either 1, 3, 7, 21, 33, 77 or 231. Upon inspection, the only possibility is 1.
Let H be...
Show that if H is a normal subgroup of G of prime index p, then for all subgroups K of G, either
(i) K is a subgroup of H, or
(ii) G = HK and |K : K intersect H| = p.
Homework Statement
Let a and b be integers
(a) Prove that aZ + bZ is a subgroup of Z+
(b) prove that a and b+7a generate aZ + bZ
Homework Equations
Z is the set of all integersThe Attempt at a Solution
(a)
In order for something to be a subgroup it must satisfy the following 3...
Homework Statement
Let G be a group and let H,K be subgroups of G.
Assume that H and K are Abelian. Let L=(H-union-K) be the subgroup of G generated by the set H-union-K. Show that H-intersect-K is a normal subgroup of L.
The Attempt at a Solution
How do i start this?
Homework Statement
Suppose that N and M are two normal subgroups of G and that N and M share only the identity element. Show that for any n in N and m in M, nm = mn.
The attempt at a solution
I basically have to show that NM is abelian. Since N and M are normal, it follows that
nm =...
A subgroup H of a group G is fully invariant if t(H)<=H for every endomorphism t of G. Let G is finite p-group has a fully invariant subgroup of order d for every d dividing |G|. What is the structure of G ?