If G acts on a set A transitively, then show that any subgroup of G also acts on 4 by the restriction of the action of G on A Further, then show that the action of l on A may not be transitive.

Q: Suppose that H is a subgroup of G and let G act by conjugation on the set X = {gHg-1 : g ∈ G}.…

A: Given that,H is a subgroup of G and G act by conjunction on the set

Q: Show that if H and K are subgroups of G then so is H ∩ K.

A: Given that H and K are subgroup of group G. We have to show that H∩K is a subgroup of group G.…

Q: Let G = . The smallest subgroup of G containing a^8 and a^12 is generated by * a^12 a^4 a^6 a^2…

A: Smallest subgroup a^m & a^n  is a^gcd(m,n) = a^gcd(8,12) = a^4 So option 2nd are correct

Q: ) If G is a group and X is a G-set, then the subset (g in G: for all x

A: To explain the given statement is true or false as,

Q: 3) Prove that if A and B are subsets of G with A C B then Cc(B) is a subgroup of CG(A).

A:

Q: 10. Let A be a subgroup of G, and let B be a subgroup of H. Show that A×B is a subgroup of G×

A:

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

A: Given: 2Z is a subgroup of (Z,+). We have to find the right coset of -5+2Z.

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -3 + 2Z contains the…

A:

Q: 5. Let p and q be two prime numbers, and let G be a group of order pq. Show that every proper…

A: We have to prove that: Every proper subgroup of G is cyclic. Where order of G is pq and p , q are…

Q: (a) Prove that if K is a subgroup of G and L is a subgroup of H, then K x L is a subgroup of G x H.

A: The detailed solution of (a) is as follows below:

Q: If |G| = 36 and G is non-Abelian, prove that G has more than oneSylow 2-subgroup or more than one…

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Q: Let A and B be subgroups of G such that AB = BA. Then AB is a subgroup of G.. Select one: True False

A:

Q: If H is a proper subgroup of L and L is a proper subgroup of G. If |G| = 1250 and |L| = 125, list…

A:

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

A:

Q: Prove If S1 and S2 are subgroups of G, then S1 intersection  S2 is a subgroup of G.

A: Let S1 and S2 are two subgroups Then if x, y E S1 or S2 .xy E S1 or S2 And V x E S1 or S2 Then x-1 E…

Q: a) Make a b) Show that if with |K| list of all subgroups of Q8 K is is a subgroup of Qs. k is…

A:

Q: Suppose that H is a proper subgroup of Z under addition and that Hcontains 12, 30, and 54. What are…

A: It is given that H is a proper subgroup of Z under addition and that H contains 12, 30 and 54.

Q: There is a group G and subgroups A and B of orders 4 and 6 respectively such that A N B has two…

A:

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

A:

Q: Let G = (a) and |a| = 24. List all generators for the subgroup of order 8.

A:

Q: If N is a normal subgroup of a group G, and if every member of N and G/N have a finite order, prove…

A: Given: If N is a normal subgroup of a group G, and if every member of N and GN have a finite order…

Q: prove That :- let H and K be subgroups of agroupG of the m is ormal a HK is subgroup of G →1f one

A: Subgroup of a group G

Q: 4. If H is a subgroup of G, then show that the set W = ngHg¹ is a normal 9€G subgroup of G.

A: Given That : H is a subgroup of G  To Show: The set W=∩g∈GgHg-1 is a normal Subgroup.

Q: Prove that S, is isomorphic to a subgroup of An+2-

A: An even permutation can be obtained as the composition of an even number and only an even number of…

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: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

A: 2Z ={ .........  , -8, -6 , -4 , -2 , 0 , 2, 4, 6 , 8, ....}

Q: Show that S = SU(2) contains a subgroup isomorphic to S'.

A: Let's define S1 as the set { (x,y)∣ x2+y2 = 1 } ⊂ R2 we may think of S3 as S3={ (a,b) ∈ C2:…

Q: Let K be a subgroup of G and let H = {g : gKg^-1} = K. Show that H is a subgroup of G

A: Let K be a subgroup of G and H be defined as H = { g : gKg-1 = K }. Then, we have to show that H is…

Q: If H and K are subgroups of a group G, prove that ANB is a subgroup of G.

A: GIVEN if H and K are the subgroup of a G, prove that  A∩B  is a subgroup of G

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -3 + 2Z contains the…

A:

Q: Every subgroup of a group G is normal * False True

A:

Q: If A is a group and B is a subgroup of A. Prove that the right cosets of B partitions A

A: Given : A be any group and B be any subgroup of A. To prove : The right cosets of B partitions A.

Q: Show that the subgroup generated by any two distinct elements of order 2 in S3 is all of S3.

A: Given that, S3 is a symmetric group of permutations. Thus, S3 has 6 elements. By using Lagrange's…

Q: Question # If G acts on a set A tronsitivelj, show that any Subgroup H and G also acts on…

A:

Q: Every subset of every group is a subgroup under the induced operation. True or False then why

A: True or FalseEvery subset of every group is a subgroup under the induced operation.

Q: Let H be a Sylow p-subgroup of G. Prove that H is the only Sylowp-subgroup of G contained in N(H).

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Q: | Suppose that p: U15 → U15 is an automorphism. Define H = {x € U15 |¢(x) = x¬1}. Which of the…

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Q: If a simple group G has a subgroup K that is a normal subgroup oftwo distinct maximal subgroups,…

A: Here given G is simple group and K is a normal subgroup of G. Then use the definition of simple…

Q: Find a subgroup of order 4 in U(1000).

A:

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

A: 7 is the correct answer.

Q: Prove that any group of order 75 can have at most one subgroup of order 25.

A:

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -6 + 2Z contains the…

A: 10 is the element in the right coset.

Q: 1. If H be a subgroup of group G, then the relation on G defined by a ~ b if and only if abe H, for…

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Q: Prove that the intersection of two subgroups of G is itself a subgroup of G.

A: Suppose that, H1 and H2 be the sub groups of G. Also, let H=H1∩H2 . Consider one step group test:…

Q: 2) Let (G, *) be a group and H, K be subgroups in G. Prove that subset H * K is a subgroup if and…

A:

Q: Determine if B is a subgroup of A

A: Introduction: Subgroup is a part of a group. More precisely, a subgroup is a non-empty subset of a…

Q: Let H be a subgroup of Sn. (a) Show that either all the permutations in H are even, or else half the…

A:

Q: Let a and b be nonidentity elements of different orders in a group Gof order 155. Prove that the…

A: Given that Let a and b be nonidentity elements of different orders in a group G of order 155. Note…