Normal subgroup Definition and 49 Threads

In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G is normal in G if and only if gng−1 ∈ N for all g ∈ G and n ∈ N. The usual notation for this relation is



N
◃
G


{\displaystyle N\triangleleft G}
.
Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of G are precisely the kernels of group homomorphisms with domain G, which means that they can be used to internally classify those homomorphisms.
Évariste Galois was the first to realize the importance of the existence of normal subgroups.

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  1. R

    Show N is a normal subgroup and G/N has finite element

    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##...
  2. Alex Langevub

    An exercise with the third isomorphism theorem in group theory

    Homework Statement Let ##G## be a group. Let ##H \triangleleft G## and ##K \leq G## such that ##H\subseteq K##. a) Show that ##K\triangleleft G## iff ##K/H \triangleleft G/H## b) Suppose that ##K/H \triangleleft G/H##. Show that ##(G/H)/(K/H) \simeq G/K## Homework Equations The three...
  3. Mr Davis 97

    Normal subgroup generated by a subset A

    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= \{...
  4. M

    MHB The extension is Galois iff H_i is a normal subgroup of H_{i-1}

    Hey! :o 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...
  5. S

    I What is the meaning of N(H) in subgroup notation?

    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!
  6. S

    I Why only normal subgroup is used to obtain group quotient

    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...
  7. M

    MHB Show that it is a normal subgroup of S4

    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)...
  8. M

    MHB Proof: Every Subgroup of Cyclic $H$ is Normal in $G$

    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 $$ghg^{-1}=h\in H, \ \forall g \in G \text{ and } \forall h\in H \tag 1$$...
  9. J

    Show group equivalence relation associated with normal subgroup

    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...
  10. Q

    Show that a normal subgroup <S> is equal to <T>

    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...
  11. R

    Finding a normal subgroup H of Zmn of order m

    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...
  12. TrickyDicky

    SU(2) as a normal subgroup of SL(2, C)

    SU(2) matrices act isometrically on the Riemann sphere with the chordal metric. At the same time the group of automorphisms of the Riemann sphere is isomorphic to the group SL(2, C) of isometries of H 3(hyperbolic space) i.e. every orientation-preserving isometry of H 3 gives rise to a Möbius...
  13. J

    Proving N is a Normal Subgroup of G

    Homework Statement N is a normal subgroup of G if aNa^-1 is a subset of N for all elements a contained in G. Prove that in that case aNa^-1 = N. Homework Equations The Attempt at a Solution Given: N is a normal subgroup of G if aNa^-1 is a subset of N for all elements a contained...
  14. W

    Embedding Group as a Normal Subgroup

    Hi, let G be any group . Is there a way of embedding G in some other group H so that G is normal in H, _other_ than by using the embedding: G -->G x G' , for some group G'? I assume this is easier if G is Abelian and is embedded in an Abelian group. Is there a way of doing this in...
  15. G

    Using generators to check for a normal subgroup

    Homework Statement Perhaps I should say first that this question stems from an attempt to show that in the group \langle x,y|x^7=y^3=1,yxy^{-1}=x^2 \rangle,\ \langle x \rangle is a normal subgroup. Let G be a group with a subgroup H . Let G be generated by A\subseteq G . Suppose...
  16. E

    Is A3 a Normal Subgroup of S3?

    H=A3= {(1),(1 2 3),(1 3 2)} and G=S3 ={ (1),(1 2 3),(1 3 2),(1 2 ),(1 3),(1 2 3) } Is H is normal subgroup of G ? I try g=(1 2 3 ) for gH=Hg but gH≠Hg for all g ε G.In this situation,H is normal subgroup pf G?
  17. O

    Normal subgroup with prime index

    Homework Statement Prove that if p is a prime and G is a group of order p^a for some a in Z+, then every subgroup of index p is normal in G. Homework Equations We know the order of H is p^(a-1). H is a maximal subgroup, if that matters. The Attempt at a Solution Suppose H≤G and...
  18. L

    Normal Subgroup Conjugate of H by element

    Homework Statement Let H be a subgroup of group G. Then H \unlhd G \Leftrightarrow xHx^{-1}=H \forall x\in G \Leftrightarrow xH=Hx \forall x\in G \Leftrightarrow xHx^{-1}=Hxx^{-1} \forall x\in G \Leftrightarrow xHx^{-1}=HxHx^{-1}=H \forall x\in G...
  19. B

    Is a Subgroup of a Normal Subgroup Always Normal?

    If N is normal in G, is its normal quotient (factor) group G/N normal as well? And does it imply that any subgroup of G/N which has a form H/N is normal as well? Is a subgroup of a normal subgrp normal as well. It appears that not always. Thanks
  20. A

    Prove that N is a normal subgroup

    Homework Statement If H is any subgroup of G and N={\cap_{a\in G} a^{-1}Ha}, prove that N is a normal subgroup of G. The Attempt at a Solution Is this statement true? \forall n: n \in N \implies \exists h \in H : n=a^{-1}ha The theorem looks intuitively true, but I don't know how to...
  21. V

    Prove that a group of order 42 has a nontrivial normal subgroup

    Homework Statement Prove that a group of order 42 has a nontrivial normal subgroup Homework Equations We are supposed to use Cauchys Theorem to solve the problem We are not allowed to use any of Sylows Theorems The Attempt at a Solution By using Cauchys Theorem i know there...
  22. N

    Is H a Normal Subgroup If Every Left Coset Equals a Right Coset?

    Let G be a group and H be a subgroup of G. If every left coset xH, where x in G, is equal to a right coset Hy, for some y in G, prove H is normal subgroup. Help?
  23. C

    Is S7 x {0} a Maximal Normal Subgroup of S7 x Z7?

    1. Is the group S7 X {0} a maximal normal subgroup of the product group S7 X Z7 ? 2. No relevant equations 3. That kinda is my answer, original question was asking about S7 X Z7
  24. L

    Reducing Normal Subgroup Calcs: Finite Groups Only?

    My abstract algebra book is talking about reducing the calculations involved in determining whether a subgroup is normal. It says: If N is a subgroup of a group G, then N is normal iff for all g in G, gN(g^-1) [the conjugate of N by g] = N. If one has a set of generators for N, it suffices...
  25. T

    Correspondence Theorem for Normal Subgroups in Groups of Order 168

    Homework Statement Show that if G is a group of order 168 that has a normal subgroup of order 4, then G has a normal subgroup of order 28. Homework Equations The Attempt at a Solution Let H be a normal subgroup of order 4. Then |G/H|=42=2*3*7, so then G?N has a unique, and...
  26. T

    Prove Intersection of Subgroups of G is Normal Subgroup

    Homework Statement Suppose that a group G has a subgroup of order n. Prove that the intersection of all subgroups of G of order n is a normal subgroup of G. Homework Equations The Attempt at a Solution I know that I need to do the following: Let S be the set of all subgroups of...
  27. 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!
  28. 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...
  29. 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...
  30. 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...
  31. 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...
  32. 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.
  33. 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...
  34. 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...
  35. 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...
  36. 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...
  37. 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.
  38. 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?
  39. T

    Normal Subgroup Equality: Closure of a Group?

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

    Normalizer &amp; normal subgroup related

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

    Inner Automorphisms as a Normal Subgroup

    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...
  42. E

    What Defines the Smallest Normal Subgroup Containing a Subset?

    [SOLVED] smallest normal subgroup Homework Statement Given any subset S of a group G, show that it makes sense to speak of the smallest normal subgroup that contains S. Hint: Use the fact that an intersection of normal subgroups of a group G is again a normal subgroup of G.Homework Equations...
  43. A

    Normal subgroup of prime order in the center

    Homework Statement Let H be a normal subgroup of prime order p in a finite group G. Suppose that p is the smallest prime dividing |G|. Prove that H is in the center Z(G). Homework Equations the Class Equation? Sylow theorems are in the next section, so presumably this is to be done without...
  44. M

    An ideal in a ring as 'analogous' to a normal subgroup of a group, but

    Hello all. I am in need of a quick clarification. A text I am reading describes an ideal in a ring as 'analogous' to a normal subgroup of a group but there appears to be a slight difference in structure in that a member of the underlying additive group from which the ideal is formed operates...
  45. V

    How can the normality of a subgroup be proven when the group has a finite index?

    Having trouble with a couple of algebra questions and would really appreciate any hints or pointers. 1. A is a subgroup of group G with a finite index. Show that N = \bigcap_{x \in G}x^{-1}Ax is a normal subgroup of finite index in G. I'm able to show that N is a subgroup of G by applying the...
  46. quasar987

    What Defines a Normal Subgroup in Group Theory?

    My professor of topology gave us a quick overview of the group theory results we will be needing later and among the things he said, is that a normal subgroup of a group G is a subgroup H such that for all x in G, xHx^{-1}=H. Is this correct? The wiki article seems to indicate that equality...
  47. A

    Characters of Normal Subgroup of Index 2

    Hi There, Ok, I'm new to this so I'm sorry if this is abit warbled!... We have a normal subgroup N of a finite group H such that [H:N]=2 We have a chatacter Chi belonging to the irreducible characters of H, Irr(H) , which is zero on H\N. I have already shown that Chi restricted to N =...
  48. A

    Are Sylow subgroups unique in groups with specific conditions?

    Suppose H and K are subgroups of G with H normal in K, |H||K| = |G|, and the intersection of H and K being identity. Then HK = G. Since HK is the union of hK for all h in H and since hK = h'K iff h = h', wouldn't the set of cosets of K be {hK : h in H}? Also, wouldn't this form a group...
  49. J

    Proof of Normal Subgroup Thm- Alternative Strategies

    Can anyone come up with an alternative proof of the following? If H, a subgroup of G, has index [G:H]=p where p is the smallest prime dividing |G|, the H is normal in G. I'm already aware of one proof, given here http://www.math.rochester.edu/courses/236H/home/hw8sol.pdf (page 3 -...
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