Recent content by mykayla10

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    Finding a Function from (G x K) to (G/H) x (K/L) for First Isomorphism Theorem

    Ok, so that makes sense. Now I am lost as to what the two phis should look like. phi1: G ->G/H and phi2: K -> K/L. However, I don't know what these look like. I am so bad at proofs.
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    Finding a Function from (G x K) to (G/H) x (K/L) for First Isomorphism Theorem

    you are allowed to use two different phis??
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    Finding a Function from (G x K) to (G/H) x (K/L) for First Isomorphism Theorem

    It says that L is a normal subgroup of K.
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    Finding a Function from (G x K) to (G/H) x (K/L) for First Isomorphism Theorem

    Homework Statement Suppose H is a normal subgroup G and L is a subgroup of K. Then (G x K)/(H x L) is isomorphic to (G/H) x (K/L) Homework Equations The Attempt at a Solution I know that I have to use the First Isomorphism Theorem, but in order to do that I need some function phi...
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    Proof of Aut(G): ϕ(Z(G))= Z(G)

    Homework Statement For every ϕ in Aut(G), ϕ(Z(G))= Z(G). Homework Equations Z(G):={g in G| gh=hg for all h in G} The Attempt at a Solution I haven't made too much progress on this one. I know that if I let g be an element of Z(G) that I need to prove that For every ϕ(g) is also...
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    Proving |H intersect K| = q for subgroup H and K in G of order pqr.

    is there any way I can eliminate |H intersect K|=1 by assuming it does and finding a contradiction? I am not sure I am at the ability level to do what you suggested.
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    Proving |H intersect K| = q for subgroup H and K in G of order pqr.

    Homework Statement Suppose |G| = pqr where p, q, and r are distinct primes. If H is a subgroup of G and K is a subgroup of G with |H| = pq and |K| = qr, then |H intersect K| = q. Homework Equations NA The Attempt at a Solution I have so far: Let a be an element of H intersect K...