Subgroup Definition and 276 Threads

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.

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...

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...

hkhk if G= Z4 x Z what would be the torsion group T(G)? and what is the factor group of G/ T(G) ?

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...

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 ?

I need a proof of any subgroup of S_n which is isomorphic to S_(n-1) fixes a point in {1, 2,..., n} unless n=6.

Does every normal subgroup equal to the normal closure of some set of a group?

Prove that if (H,o) and (K,o) are subgroups of a group (G,o), then (H \cap K,o) is a subgroup of (G,o). Proof: The identity e of G is in H and K, so e \in H\capK and H\capK is not empty. Assume j,k \in H\capK. Thus jk^{-1} is in H and K, since j and k are in H and K. Therefore, jk^{-1}...

I'm supposed to show that a subgroup of a solvable group is solvable. (I am using the Fraleigh Abstract Algebra book and the given definition of a solvable group is a group which has a COMPOSITION series in which each of the factor groups is abelian. In other books I have looked at a solvable...

Homework Statement Recall that a subgroup N of a group G is called characteristic if f(N) = N for all automorphisms f of G. If N is a characteristic subgroup of G, show that N is a normal subgroup of G. The attempt at a solution I must show that if g is in G, then gN = Ng. Let n be in N...

1) Let X be anon empty subset of a group G .prove that there is a smallest normal subgroup of G containing X ii)what do we call the smallest normal subgroup of G containing X

Homework Statement By considering the vertices of the pentagon, show that D5 is isomorphic to a subgroup of S5. Write all permutations corresponding to the elements of D5 under this isomorphism. The Attempt at a Solution To show isomorphic, need to find a function f: D5->S5, where...

Homework Statement Consider the group D4 (rigid motions of a square) as a subgroup of S4 by using permutations of vertices. Identify all the even permutations and show that they form a subgroup of D4. The Attempt at a Solution I think I have the permutations of correct. They are...

Homework Statement Let G1, G2 be groups with subgroups H1,H2. Show that [{x1,x2) | x1 element of H1, x2 element of H2} is a subgroup of the direct product of G1 X G2 The Attempt at a Solution I'm not sure how to begin solving this problem.

I'm trying to prove that GL(2,p^n) has a cyclic subgroup of order p^{2n} - 1. This should be generated by \left( \begin{array}{cc} 0 & 1 \\ -\lambda & -\mu \end{array} \right) where X^2 + \mu X + \lambda is a polynomial over F_{p^n} such that one of its roots has multiplicative order...

[SOLVED] Conjugates in the normalizer of a p-Sylow subgroup Homework Statement Let P be a p-Sylow subgroup of G and suppose that a,b lie in Z(P), the center of P, and that a, b are conjugate in G. Prove that they are conjugate in N(P), the normalizer of P (also called stablilizer in other...

Homework Statement Let M and N be normal subgroups of G, and suppose that the identity is the only element in both M and N. Prove that G is isomorphic to a subgroup of the product G/M\times G/N Homework Equations Up until now, we've dealt with isomorphism, homomorphisms, automorphisms...

[SOLVED] Normalizer &amp; normal subgroup related Let H \subset G. Why is H a normal subgroup of its own normalizer in G?

Homework Statement Let G be a group. We showed in class that the permutations of G which send products to products form a subgroup Aut(G) inside all the permutations. Furthermore, the mappings of the form \sigma_b(g)=bgb^{-1} form a subgroup inside Aut(G) called the inner automorphisms and...

Homework Statement Please confirm that the center of a group always contains the commutator subgroup. I am pretty sure its true. Homework Equations The Attempt at a Solution