Let H be the subgroup (10) of Z15. (i) What's the order of H? (ii) What's the number of left cosets of H in Z15?

Q: . Let H be a subgroup of R*, the group of nonzero real numbers un- der multiplication. If R* C H C…

A: H be a subgroup of R*, the group of nonzero real numbers under multiplication. R+⊆ H ⊆ R*. To prove:…

Q: Prove that a group of order 375 has a subgroup of order 15.

A:

Q: Prove that SL,(R) is a normal subgroup of GL,(R).

A: Let G=GL(n,R) be the general linear group of degree n, that is, the group of all n×n invertible…

Q: 2) Let H be a normal subgroup of G. If| H|-2. Prove that H is contained in the center Z(G) of G.

A:

Q: Prove that H x {1} and {1} x K are normal subgroups of H x K, that these subgroups general H x K,…

A:

Q: 11. Find the cyclic subgroup of D4 generated by µp². What is the order of this subgroup?

A:

Q: 5. Find the right cosets of the subgroup H in G for H = {(0,0), (1,0), (2,0)} in Z3 × Z2.

A:

Q: Let H be a subgroup of a group G, S {Hx:xe G). nen prove that there is a homomorphism of G onto A(S)…

A:

Q: Let G be a group of order 24. Show that G is solvable

A: Let G be a group of order 24. Show that G is solvable

Q: Find the three Sylow 2-subgroups of D12 using its subgroup lattice below. E of G Let r v E G…

A:

Q: a. Prove or Disprove. If H is an abelian normal subgroup of G then H be contained in Z(G).

A:

Q: If H is a normal subgroup of G and |H| = 2, prove that H is containedin the center of G.

A:

Q: In Z24, list all generators for the subgroup of order 8.  Let G = and let |a| = 24.  List all…

A:

Q: Ql: Prove that (Q\{0},x) is a subgroup of (R\{0},x).

A:

Q: Let H be the subgroup of S3 generated by the transposition (12). That is, H = ((12)) Prove that H is…

A: We know that S3=1, 12, 13, 23, 123, 132. Giventhat H=12 is a subgroup of S3. H=1, 12We have to show…

Q: If N is a normal subgroup of G and G/N=m , show that xmN forall x in G.

A:

Q: Let G be the subgroup of GL3(Z2) defined by the set 100 a 10 b C 1 that a, b, c Z₂. Show that G is…

A: Given: G is the subgroup of GL3ℤ2 which is defined by the set of matrix 100a10bc1 where a, b, c∈ℤ2 .…

Q: 1. Show that H={[0], [2], [4]} is a subgroup of a group (Z6+6). Obtain all the distinct left cosets…

A: Given that H=0,2,4 and let G=ℤ6,+6.

Q: 16* Find an explicit epimorphism from S5 onto a group of order 2

A: To construct an explicit homomorphism from S5 (the symmetric group on 5 symbols) which is onto the…

Q: H be a subgroup of G.

A: We have to find out the truth value of the given statements. It is given that H is a subgroup of G.…

Q: Prove that A5 is the only subgroup of S5 of order 60.

A:

Q: KE Syl-(G). Prove that (a). HG and KG. (U). G has a cyclic subgroup of order 77. Syl(G),

A:

Q: Let n > 2 be an integer. Prove that An is a normal subgroup of Sn.

A: In abstract algebra, a normal subgroup is a subgroup that is invariant under conjugation by members…

Q: Let |G|=pq, where p and q are prime.  If G has only one subgroup of order p and only one of order q,…

A: Given |G|=pq and G has only one subgroup of order p and only one of order q. To prove that G is…

Q: Let M and N be normal subgroups of G. Show that MN is also a normal subgroup of G

A: It is given that M and N are normal subgroups of G. implies that,

Q: In Z24, list all generators for the subgroup of order 8. Let G = <a>and let |a| = 24. List all…

A:

Q: 17*. Find an explicit epimorphism from A5 onto a group of order 3

A: Epimorphism: A homomorphism which is surjective is called Epimorphism.

Q: Let H and K be normal subgroups of a group G such at HCK, show that K/H is a normal subgroup of G/H.

A:

Q: let H be a normal subgroup of G and let a belong to G . if the element aH has order 3 in the group…

A:

Q: 4. a) Let H be the set of elements [a b] of G of GL(2,R) such that ab- bc = 1. Show that H is a…

A:

Q: let H be a normal subgroup of G and let a belong to G. if th element aH has order 3 in the group G/H…

A: H is normal subgroup of G. And a belongs to G.  O( aH) = 3 in G/H and O(aH) in G/H divides  O(a) in…

Q: is a subgroup of Z_18 of order: 18 1 4 3

A:

Q: 6. (b) For each normal subgroup H of Dg, find the isomorphism type of its corresponding quotient…

A: First consider the trivial normal subgroup D8. The quotient group D8D8=D8 and hence it is isomorphic…

Q: Let H be a subgroup of G of index 2. Prove that H is a normal sub-group of G.

A: the prove is given below...

Q: Find all generators of the subgroup of Z/60Z with order 12.

A: Generators of the group

Q: Use the three Sylow Theorems to prove that no group of order 45 is simple.

A:

Q: Q3: (A) Prove that 1. There is no simple group of order 200. 2. Every group of index 2 is normal.

A: Sol1:- Let G be a group of order 200  i.e O(G) = 200                = 5² × 8. G contains k Sylows…

Q: Prove that group A4 has no subgroups of order

A: Topic- sets

Q: Prove that H= { |ne Z} is a subgroup of GL2(R) under multiplication.

A:

Q: Use the left regular representation of the quaternion group Q8 to produce two elements of Sg which…

A: Fix the labelling of Q8 , Take elements 1, 2, 3, 4,  5, 6, 7, 8 are 1, -1, i, -i, j, -j, k, -k…

Q: 2. A Sylow 3-subgroup of a group of order 54 has order

A:

Q: 8. Is Zg isomorphic to D4? What about Z4 and D4? Can you find a subgroup of D4 isomorphic to Z4?

A: Now we have to answer the above question .

Q: Find the center and the commutator subgroup of S3 x Z12-

A: Solution

Q: If H is a subgroup of G such that [G : H] = 2, then show that H is a normal subgroup of G.

A: Suppose H≤G such that [G:H] = 2. Thus H has two left cosets (and two right cosets) in G.

Q: Obtain al the Sylow p-subgroups of (Z/2Z) X S3

A:

Q: Prove that every group of order 78 has a normal subgroup of order 39.

A:

Q: Let G = Z9 X Z12 X Z16- (a) Find all subgroups of G of order 144.

A:

Q: Let H be a subgroup of G such that x^2 ∈ H for all x ∈ G, then show that H is a normal subgroup of…

A: H = {x² : x ∈ G} And, H < G

Q: Find the Galois group of the polynomial r-1.

A:

Q: 6. Let G be a group of order p², where p is a prime. Show that G must have a subgroup of order p.

A: