What is Subgroup: Definition and 290 Discussions

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.

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  1. E

    Is There a Normal Subgroup K in Groups G and H with Index (G:K) ≤ n!?

    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!
  2. B

    Not a specific problem, just some help with the one step subgroup test

    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
  3. C

    BRS: Subgroup lattice of a Permutation Group via GAP

    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...
  4. K

    Is there a Program that Identifies and Names Subgroups of a Defined Group?

    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...
  5. T

    Abstract Algebra: Proving whether H is a subgroup.

    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...
  6. P

    Normal Cyclic Subgroup in A_4: Proving Normality and Identifying Elements

    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...
  7. P

    Showing that g^-1 H g is a subgroup

    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...
  8. T

    Alternating group is the unique subgroup of index 2 in Sn?

    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...
  9. quasar987

    Elementary property of maximal compact subgroup

    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...
  10. H

    Subgroup Generated by {4,6} in Z12

    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...
  11. 3

    Isomorphism between G and Z x Z_2 if G has a normal subgroup isomorphic to Z_2

    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...
  12. V

    How to show SU(n) is a normal subgroup of U(n)

    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...
  13. 3

    The size of the orbits of a finite normal subgroup

    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...
  14. 3

    Orbits of a normal subgroup of a finite group

    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...
  15. P

    Abstract Algebra: Commutative Subgroup

    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...
  16. J

    Closure of an abelian subgroup

    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...
  17. B

    1 maximal subgroup -> prime order

    '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...
  18. M

    Proving Normal Subgroup of S4 in Alternative Ways

    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.
  19. C

    Counting p-Sylow Subgroups for S4 and Z/5 with p=2, 3, and 5

    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...
  20. A

    Non-abelian subgroup of size 6 in A_6

    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...
  21. L

    Intersection of 2 subgroups is a subgroup?

    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...
  22. M

    Show G/\Phi(G) is Elementary Abelian p-Group

    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...
  23. C

    Group actions of subgroup of S_3 onto S_3

    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...
  24. C

    Normalizer subgroup proof proving the inverse

    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...
  25. G

    Verifying Noncyclic Abelian Subgroup of S4

    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...
  26. J

    Subgroup Theorem: Proof and Explanation

    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...
  27. mnb96

    Subgroup wth morphism into itself

    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...
  28. J

    Normal subgroup of a product of simple groups

    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...
  29. H

    Torsion group, torsion subgroup

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

    Can a Sylow Subgroup be Contained in a Sylow p-Subgroup?

    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...
  31. R

    Subgroup of D_n: Proving <f> Not Normal

    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
  32. H

    Subgroup K Normal in Dn: Proof and Examples

    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...
  33. T

    Normal subgroup; topological group

    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...
  34. H

    Solving Group Problems: |g| = 20 in G and Subgroup H = <x,y>

    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...
  35. H

    Give an example where H is not a subgroup.

    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...
  36. M

    Orthochronal subgroup of the Lorentz group

    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...
  37. M

    Mobius Inversion, finite subgroup

    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)...
  38. M

    What is the Significance of the Orbit of P in Sylow's Theorems?

    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.)
  39. Z

    Exploring the Properties of Subgroup <[4]> in Z13

    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...
  40. E

    Subgroups of a Cyclic Normal Subgroup Are Normal

    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...
  41. malawi_glenn

    Lie Subgroup - Cartan's Theorem

    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...
  42. E

    Is HK a Subgroup of S_5? - Homework Statement & Equations

    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...
  43. E

    Is the 11-Sylow subgroup of G in the center of G?

    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...
  44. D

    Normal Subgroup of Prime Index: Properties

    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.
  45. S

    Proving Closure and Identity of aZ + bZ as a Subgroup of Z+

    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...
  46. F

    Proving Normal Subgroup of Abelian Groups

    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?
  47. E

    Proving Commutativity in Normal Subgroups with Abelian Subgroup Problem

    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 =...
  48. R

    A question of fully invariant subgroup

    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 ?
  49. D

    Subgroup of Sym(n) Isomorphic to S_(n-1) Except n=6

    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.
  50. T

    Normal Subgroup Equality: Closure of a Group?

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