assignments

Assignment 1

• In class we met the group \( C \), the symmetry group of the cube (not allowing reflections). Either prove or disprove that it is isomorphic to \( S_4 \) the symmetric group on four symbols.

• Three parts (the third is considerably trickier than the first two):
  • Show that \( (gh) ^2 = g^2h^2 \) if and only if \( gh = hg \).
  • Show that \( (gh)^{-1} = g^{-1}h^{-1} \) if and only if \(gh=hg\).
  • Show that if there exists a number \(m \) such that the equation \( (gh)^n = g^nh^n \) holds for \(n=m,\) \(n=m+1\) and \(n=m+2\) then \( gh = hg \).

• Let G= \(( -c , c) \) be an open interval of the real line. Define \( x \circ y = \frac{x+y}{1+xy/c^2} \). Is \( G \) a group? [Hint: Tricky by direct calculation. You might like to consider \( f(x) = c \tanh(x) \) if you know some hyperbolic trig, or maybe you know where in Physics this pops up?]

• Let G be a set with binary operation \( \circ \) for which our usual closure and associative axioms hold.
  • If in addition the following two axioms hold, is G necessarily a group?
    • There exists \( e \in G \) such that \(e \circ g = g\) for all \( g \in G \).
    • For any \( g \in G \), there exists \( h \in G \) such that \( h \circ g = e\).
  • Same question, but with these two additional axioms instead:
    • There exists \( e \in G \) such that \(e \circ g = g\) for all \( g \in G \).
    • For any \( g \in G \), there exists \( h \in G \) such that \( g \circ h = e\).

Assignment 2

• Show that a finite group \( G \) contains an involution (i.e. an element of order 2) if and only if \( |G| \) is even.

• Suppose \(G\) has a subgroup \(H\) of index 2. Show that \(H\) is normal in \( G\).

• Find subgroups \(H\) and \(J\) of \(S_3\) such that \(HJ\) is not a subgroup of \( S_3\).

• Find all homomorphisms from \( \mathbb{Z}_7 \) to \( S_5 \).

• In response to last time's axiom-related question, I mentioned in class that the following is sufficient to define a group. Let \(G\) be a set with a binary operation \(\circ\) and a unary operation \( ' \) (think group multiplication/addition and inverse respectively, and being an "operation" includes closure) such that \( (a \circ b) \circ c = (a \circ d) \circ f \) implies that \( b = d \circ (f \circ c') \). Can you show that \(G\) is indeed a group? [Disclaimer: I haven't looked at the proof or tried it myself yet; I don't know how hard this is.]

• A Latin square of order \(n\) is an \( n \times n\) array of \( n\) symbols such that each symbol occurs once in each row and once in each column. Show that the Cayley table of a finite group is a Latin square (once you've removed the border).

• Same extra blurb as last time: As we discussed, also choose some questions to answer from the book you're working from (or elsewhere). In addition, a few bits above are definitely quite hard, and there's a chance they might not line up well with what you've read. So also feel free to say where you've struggled and sub other questions in. More generally, commentary on your solutions is welcome.

Assignment 3

• Let \(R\) be a ring in which every element satisfies \( x^2 = x \). Show that \(x+x=0\) for all \( x \in R \). Show that \(R\) is commutative.

• Let \( I \) be an ideal in a commutative ring \( R\). Prove that \(I[x]\) is an ideal in \(R [x] \) (where the notation \( S[x] \) means the ring of one-variable polynomials in \(x\) with coefficients in the ring \(S\) ). Describe \( R[x]/I[x] \).

• Look up and read about Euclid's Algorithm and the proof it gives that for integers \( a \) and \( b \) we can find integers \( x \) and \( y \) such that \( \gcd(a,b) = ax+by\). Let \(p\) be a prime number and let \(a\) be a positive integer such that \( a\not\equiv 0 \mod p\). Show that there exists \(c\) such that \(ca\equiv 1 \mod p\).

• (For Friday) Find an interesting ideal of \( \mathbb{R}[x] \).

• Usual extra blurb: As we discussed, also choose some questions to answer from the book you're working from (or elsewhere). In addition, a few bits above are definitely quite hard, and there's a chance they might not line up well with what you've read. So also feel free to say where you've struggled and sub other questions in. More generally, commentary on your solutions is welcome.

Assignment 4

• Do some extra problems, as usual (except I've put it first to maybe encourage more of them).

• Find the greatest common divisor of the real polynomials \( f(x) = x^2 + 3x + 2 \) and \( g(x) = x^5 + 2x^4 + 5x^3 + 6x + 2 \). Give a simple description of the ideal \( (f(x), g(x) \) of \( \mathbb{R}[x] \).

• Find a irreducible quadratic polynomial over \( \mathbb{Z}_3 \). Hence construct a field of order 9.

• Show that if \( F \) is a field with \( q \) elements and \( f \) is an irreducible polynomial of degree \( n \) over \( F \) then the field \( F[x]/(f) \) has \( q^n \) elements.

Final Assignment

• Let \( G \) be a group. Let \( \hat{G} = \{ (g,g) : g \in G \} \subseteq G \times G \). Show that \( \hat{G} \) is a subgroup of \( G \times G \) and that it is a normal subgroup if and only if \( G \) is abelian.
• Let \( D_{2n} \) denote the dihedral group with \( 2n \) elements and \( S_n \) denote the symmetric group on \( n \) symbols. Show that \( D_{2n} \) can be embedded in \( S_n \) (i.e. there is an isomorphism from \( D_{2n} \) to a subgroup of \( S_n \) ).

• Define addition and multiplication on the set \( \mathbb{Z} \times \mathbb{Z} \) as follows:
  • \( (a,b) + (c,d) = (a+c, b+d) \)
  • \( (a,b)(c,d) = (ac+2bd, ad+bc) \)
• Show that this gives a commutative ring with identity.

• An element \( a \) of a ring \( R \) is called nilpotent if \( a^n = 0 \) for some positive integer \( n\). Show that the set \( N \) of all nilpotent elements in a commutative ring \(R\) is an ideal of \( R \). Describe all of the nilpotent elements of \( R/N \).

• Show that if \( F \) is a field with \( q \) elements and \( f \) is an irreducible polynomial of degree \( n \) over \( F \) then the field \( F[x]/(f) \) has \( q^n \) elements. (Yes, this should look familiar, but I don't think that anyone quite got it completely last time around.)