Let K and H be subgroups of a finite group G with KCHCG.If[G:H] = 4 and [H:K] = 3. Then, [G:K] = 3 4 None of the choices 12 O 48

Q: 1. Let a and b be elements of a group G. Prove that if a E, then C. 2. Let a and b be elements of a…

A:

Q: . Let N be a finite group and let H be a subgroup of N. If [H| is odd and [N:H] = 2, prove that the…

A:

Q: O If a group G acts on a set S, every element of S is fixed by the identity of G. O Every group of…

A: Hello. Since your question has multiple sub-parts, we will solve first three sub-parts for you. If…

Q: 9) Let H be a subgroup of a group G and a, be G. Then a e bH if and only if O b-la e H O ba e H O…

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: Let H and K be subgroups of a finite group G with H C KC G. Prove that |G:HI |G:K| |K:H].

A:

Q: Suppose that G is a group and |G| = pnm, where p is prime andp >m. Prove that a Sylow p-subgroup…

A:

Q: If a is an element of order 8 of a group G, and = ,then one of the following is a possible value of…

A: Given that a is an element of order 8 and a4=ak

Q: Let H be a subgroup of a group G and a, be G. Then be aH if and only if None of these O ab e H O…

A: We know that b∈bH                                    (1) We know that aH = bH if and only if a-1b∈H…

Q: If G is a finite group with |G|<160 and G has subgroups of orders 1O, 16 and 20 then the order of G…

A:

Q: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if * ba EH O None of these…

A:

Q: Let N be a finite group and let H be a subgroup of N. If |H| is odd and [N:H] = 2, prove that the…

A: According to the given information, Let N be a finite group and H be a subgroup of N. H is odd and…

Q: If d divides the order of a cyclic group then this group has a subgroup of order d. Birini seçin: O…

A:

Q: Suppose H and K be subgroups of a finite group G with |G : H| = m and |G : K| = n. Prove that…

A: We use here, Tower law of subgroup which states that Let (G,∘) be a group.  Let H be a subgroup of G…

Q: For each of the following groups G and subgroups H, how many distinct left cosets of H in G are…

A: The given group is G and H≤ G. To find: How many distinct left cosets of H in G.

Q: 16. Let G be a group and let g and h be in G. Show that . - as defined in #15.

A: Given: G is a group, g,h∈G, ϕg:G→G is a function defined as ϕg(x)=gxg-1 To prove: ϕg∘ϕh=ϕgh Let x∈G…

Q: Let G be a finite group. Then G is a p-group if and only if |G| is a power of p. We leouo the

A: Given G is finite group and we have to prove G is a p-Group of and only if |G| is a power of p.

Q: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if* O ba e H O b-1a e H…

A: We will use definition of left coset

Q: Let G be a group and let H and K be subgroups of G. Prove that the intersection of H and K, H n K =…

A: 1. It is given that H∩K=x∈G|x∈H and x∈K Let  x, y∈H∩K ⇒x,y∈H and x,y∈K⇒xy-1∈H and xy-1∈K⇒xy-1∈H∩K…

Q: Let G be a finite group and a€G s.t |a|=12.if H= find all other generators of H.

A:

Q: If G is a finite group with |G|<120 and G has subgroups of orders 1O, 15 and 20 then the order of G…

A: Using Lagrange's theorem it can be written that if G is a finite group and B is a subgroup of G,…

Q: If G is a finite group, H ≤ G, the order of H divides the order of G: | H | / | G | Prove

A:

Q: Let K and H be subgroups of a finite group G with KCHCG. If [G:H] = 4 and [H:K] = 3. Then, [G:K] =…

A:

Q: 5) In each of parts (a) to (c) show that for specified group G and subgroup A of G, Cg(A) = A and…

A:

Q: Given the groups R∗ and Z, let G = R∗ ×Z. Define a binary operation ◦ on G by (a, m) ◦ (b, n) = (ab,…

A:

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

A:

Q: Suppose that a is an element of a group G. Prove that if there is some integer n, n notequalto 0,…

A: Suppose that a is an element of a group G. Prove that if there is some integer n, such that n≠0 and…

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 ano then the order of G is:…

A:

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

A:

Q: et G be a group and suppose that x E G has order n. Let d be a divisor of n. Show that G as an…

A:

Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

A:

Q: Suppose that G is a finite group and let H and K be subgroups of G. Prove that |HK| = |H||K|/|HN K|.

A:

Q: If G is a finite group with |G|<160 and G has subgroups of orders 10, 16 and 20 then the order of G…

A: Given G&lt;160 It have the subgroups of order 10 , 16 , 20 Hence G must be the common multiple of 10…

Q: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if * ba-1 E H ba E H O b-1a…

A: Q9. Third option is correct.

Q: Let H and K be subgroups of a group G. Prove that HNK is a subgroup of G.

A:

Q: If G is a finite group with IG|<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

A:

Q: If G is a finite group with IG|<160 and G has subgroups of orders 10, 16 and 20 then the order of G…

A: Given :- order of a group G... |G| &lt; 160 Also order of subgroups of group G are 10, 16 , 20. We…

Q: Let H and K be subgroups of the group G, and let a, b E G. Show that either aH n bK = Ø or else aH N…

A:

Q: 5) In each of parts (a) to (c) show that for specified group G and subgroup A of G, CG(A) = A and…

A: Given G=S3 and A=1,1 2 3, 1 3 2 The objective is to show that CGA=A, NGA=G. The definition for…

Q: If G is a finite group with IG|<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

A: Using Lagrange's theorem it can be written that if G is a finite group and A is a subgroup of G,…

Q: Let G be a group and g E G. Prove that if H is a Sylow p-group of G, then so is gHg-1

A: It is given that, G is a group and g∈G. To sow that if H is a sylow p-subgroup of G, then so is…

Q: If G is a finite group with |G|<180 andG has subgroups of orders 10, 18 and 30 then the order of G…

A: Use the fact that order of subgroup divides order of group

Q: Suppose that G is a group and |G| = pnm, where p is prime and p &gt; m. Prove that a Sylow…

A:

Q: 5. Let G be the symmetric group S3. Calculate NG(H) when H is i. the subgroup {1, (12)} ii. the…

A: The normalizer NGH of a subgroup H of a group G can be defined to be a set NGH=g∈G gHg-1=H or…

Q: 9) Let H be a subgroup of a group G and a, bE G. Then a E bH if and only if * None of these b-1a e H…

A: Second option is correct.

Q: Let K and H be subgroups of a finite group G with KCHCG.lf [G:K] = 12 and [H:K] = 3. Then, [G:H] =…

A: Let , K  and  H  be subgroups of finite group G. Also . K ⊆ H ⊆ G Here , G : K = 12  , H : K = 3 We…

Q: If G is a finite group with |G|<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

A:

Q: Suppose that G is a group such that Ord(G) = 36. The number of subgroups that G has is 4 O 12 O 18…

A: Given order of G is 36 So U(G) = {1,5,7,11,13,17,19,23,25,29,31,35} So number of elements are 12…

Q: Suppose that G is a group such that Ord(G) = 36. The number of subgroups %3D that G has is 4 О 12 О…

A: Order of a group: Let G be a group and n be the number of elements in the group. Then, order of…