# subgroup

1. ### If H proper subgroup of G then H is not isomorphic to G.

Hi: A group is ACC if every increasing sequence of normal subgroups stops. It has DCC if every decreasing sequence of normal subgroups stops. $G$ has both chain conditions if it has both ACC and DCC. The problem is this: If $G$ has both chain conditions, then there is no proper subgroup $H$ of...
2. ### Possible subgroups of an abelian finite primary subgroup.

Hi: The problem: Let G be an abelian finite group and H <= G. Then G has a subgroup isomorphic with G/H. Of course I can begin with an arbitrary group Q and find what is the structure Q must have in order to be a subgroup of G. If I find it, the problem is solved, I think. Also, we may assume...
3. ### Group R^× isomorphic to the group R?

Question: Is the group R^× isomorphic to the group R? Why? R^× = {x ∈ R|x not equal to 0} is a group with usual multiplication as group composition. R is a group with addition as group composition.Is there any subgroup of R^× isomorphic to R? What I Know: Sorry, I would have liked to show some...
4. ### (Q,+) cannot have a subgroup of index 2

I want to prove the group (Q,+) cannot have a subgroup of index 2. I assume H is the subgroup of index 2.And k1 k2 k3 are three rational number, satisfying k1 +k2 =k3 and they don't belong to H. If we can show there always exist such three number, it is easy to show the union H and k1H is not...
5. ### Do the cosets of a subgroup always form a representation of the group?

G = {g1, g2, g3, g4, g5, g6, g7, E} Lets say the cosets of a subgroup H1 of G are: {g1, g2} {g3, g4} {g5, g6} {g7, E} so G = { {g1, g2}, {g3, g4}, {g5, g6}, {g7, E} } = { a, b, c, d} so the representation { a, b, c, d} has order 4 and the subgroup H1 has 4 cosets My question is...
6. ### L_n(V,F) is a subgroup of Sv, the symmetric group of V.

Hi: In Group Theory, a book in the Schaum series, p.89, it says: "First let us define L_n(V,F) to consist of all one-to-one linear transformations of V, the vector space of dimension n over F. L_n(V,F) is included in Sv, the symmetric group of V, clearly". Then he uses this fact to prove that...

10. ### Help with the subgroup test

I am having trouble with this. say I want to prove in general a non-empty subset $H$ of $G$ is a subgroup first, I assume it is a subgroup then for all $a,b \in H$ there exists $ab^{-1} \in H$ then let $a=b$ thus $ab^{-1} = aa^{-1} =e \implies e \in H$ thus the identity is in $H$ and...
11. ### Transitive subgroup of the symmetric group Sn

Hi, I need help in proving the following statement: An abelian,transitive subgroup of the symmetric group Sn is cyclic,generated by an n-cycle. Thank's in advance
12. ### verification that subgroup is normal

Hello, we're supposed to verify that a certain subgroup X is normal, and also find a group isomorphic to $(G,*)/X$ As usually, the teacher showed us an algorithm how to do all that without much explanation, and I'm left wondering why are things the way the are. G is a group of 3x3 matrices...
13. ### number of subgroup

Suppose a group G has countable number of elements. Does G have uncountable number of distinct subgroups?
14. ### abstract algebra - subgroup question

hello everybody mathematics lover i have a question and excuse me for my language is not good if G is Finitude and A is subgroup of G then for all x and y in G ; o(AxA)=o(AyA) how can i prove this : for all g in G ; gA(g-inverse)=A ------------------------------------------------------- this...
15. ### Is Z/2Z a subgroup of Z/3Z?

Is it? \mathbb{Z} / 2 \mathbb{Z} = \{ \bar{0}, \bar{1} \} \subset \mathbb{Z} / 3 \mathbb{Z} = \{ \bar{0}, \bar{1}, \bar{2} \} It seems to be closed and there does seem to be inverses in it...? Like \bar{1} + \bar{1} = \bar{2} = \bar{0} \in \mathbb{Z} / 2 \mathbb{Z}
16. ### Show that a subgroup is normal

Let G be a group. Prove that N = \langle x^{-1} y^{-1} xy \, \, | \, \, x, y \in G \rangle is a normal subgroup of G and G/N is abelian So starting with the first part... To show something is a normal subgroup we have to show that g (x^{-1} y^{-1} xy) g^{-1} \in N for some g in G. I've been...
17. ### Normalizer of a group and its relation to a subgroup

For a) I proved the subgroup property. My problem is the example. H clearly has to be a group of some kind else the statement "H is not a subgroup of N_G(H)" is empty of meaning. So H has to be a group, but not a subgroup of G. I can't image how this could arise. Any hints? For b) I have...
18. ### Proving subgroup statements

Suppose that H and K are subgroups of a group G. Prove the following statements under additional conditions: (a). First, give an example that HK may not be a subgroup of G. (b). Prove that if H is a normal subgroup of G, then HK is a subgroup of G. (c). Prove that if both H and K are normal...
19. ### S4 subgroup of order 12

Show that S4 (the symmetric group of degree 4) has a unique subgroup of order 12. I know that A4 is that subgroup but I'm not really sure how to show that it is the unique subgroup. Help?
20. ### Help with a proof - the commutator subgroup

I need to prove the following: Let G be a group. If H<G s.t. S<H, then H\triangleleft G (S is the commutator subgroup). I just need a lead here... Thank you in advance.