# What is Subgroup: Definition and 289 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. ### I Proof for subgroup -- How prove it is a subgroup of Z^m?

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...

11. ### I Proving that a subgroup is normal

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##?
12. ### Proving Subgroup of Z/3Z

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...
13. ### Showing that preimage of a subgroup is a subgroup

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##...
14. ### Showing that a subgroup of Sym(4) is isomorphic to D_8

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...
15. ### I Using group action to prove a set is a subgroup

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...
16. ### If H is a subgroup, then H is subgroup of normalizer

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...
17. ### Show that union of ascending chain of subgroups is subgroup

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...
18. ### Showing that upper triangular matrices form a subgroup

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...
19. ### Conditions on H and K if H ∪ K is a subgroup

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...
20. ### Subgroup of Index ##n## for every ##n \in \Bbb{N}##.

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...
21. ### I Issue regarding a subgroup

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...
22. ### Subgroup of an arbitrary group

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...
23. ### MHB Proving K is a Subgroup of G: Subgroup Nesting in H and L

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?
24. ### MHB Proving N(H) is a Subgroup of G Containing H

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.
25. ### Showing that dihedral 4 is isomorphic to subgroup of permutations

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...
26. ### I Generalizing the definition of a subgroup

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...
27. ### I Definition of discrete Subgroup quick q

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...
28. ### Group theory -- show H is a subgroup of O(2)

Homework Statement Let ##R(\theta) = \left( \begin{array}{cc} \cos(\theta) & -\sin(\theta)\\ \sin(\theta)& \cos(\theta)\\ \end{array} \right) \in O(2)## represent a rotation through angle ##\theta##, and ##X(\theta) = \left( \begin{array}{cc} \cos(\theta) & \sin(\theta)\\ \sin(\theta)&...
29. ### Prove (Q+, *) is isomorphic to a proper subgroup of itself

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...

39. ### MHB Find Left Cosets of Subgroup in $\mathbb{Z}_{15}, D_4$

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...
40. ### 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...
44. ### Elements of semigroup commuting with subgroup

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...
45. ### Prove that T(G) is subgroup of G

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...
46. ### 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...
47. ### 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...
48. ### 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...
49. ### Why is the core of a subgroup contained in the subgroup?

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...
50. ### Subnormal p-Sylow Subgroup of Finite Group

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...