(b) Find all bEG with order /b| = 7 bEG. are: (c) Construct in Ga subgroup H with index.(H) = 7

Q: Let p be an odd prime number. Let Dp be the diehedral group of order 2p. Find |Aut (Dp)|.

A: Dp is the Diehedral group of order 2p, where p is an odd prime number.  We have to find AutDp.

Q: a(H n K) Let G is a group, H, K C G, and a e G. Is it the case that aH N aK? Provide a proof or…

A: According to the given information,

Q: Let a be a group element such that |a| = 48. For each part, find adivisor k of 48 such thata.…

A: Given: a be a group element such that a=48.We need to find divisors k of 48 such that the subgroups…

Q: 1. Let G be a cyclic group of order n and k❘n. Show that G has a subgroup of order k; "

A: To prove that a cyclic group ( G ) of order ( n ) has a subgroup of order ( k ) for every ( k ) that…

Q: (14) Let G be a finite cyclic group of order n with identity element e. Prove that if a € G, then cd…

A: The order of every subgroup of a finite group G is divisor of the order of G.

Q: Let G be a group. Assume a and b are elements of G and set H₁ = and H₂ =. Assume |H₁| = 21 and |H₂|…

A:

Q: describe all the elements in the cycle subgroup GL(2,R) generated by the given 2 x 2 matrix [0 -1]…

A:

Q: Let G be a group, and let a E G.  Prove that C(a) = C(a-1).

A: Given: Let G be a group and let a∈G. then we will prove C(a)=C(a-1) If C(a) be the centralizer of a…

Q: Let P₂(x) = {ax² + b | a, b e R and a #0}. If p(x) = ax² +b and g(x) = cx² + d are elements of…

A: In the given problem we have to show the the set p2(x) is a group with respect to the binary…

Q: Let a and b be elements of a group.  If |a| and |b| are relatively prime, show that intersects =…

A: Let m and n be the order of the elements a and b of a group G. Given that the orders of a and b are…

Q: (a) Let G = {e, a, . .. , aº | a10 = e} be the cyclic group of order 10. For which m, is G = (am)?…

A: As per our guideline we are supposed to answer only first asked  question. Kindly repost other…

Q: If G is a cyclic group of order n, then G is isomorphic to Zn. true or false?

A:

Q: Let G= be a finite cyclic group of order n. Suppose that n = dq where d and q are non-negative…

A:

Q: Let G be a group with IG|=209 then every proper subgroup of G is: * O Non abelian O Non cyclic O…

A: See the detailed solution below.

Q: Show that x is a generator of the cyclic group (Z3[x]/<x3 + 2x + 1>*.

A: Given: The cyclic group Z3x/x3+2x+1*We need to show that x is a generator of the given cyclic group.

Q: Let p and q be distinct odd primes. Prove that Z, is not a cyclic group.

A: The Chinese remainder theorem states that if one knows the remainders of the Euclidean division of…