Can you send a Bitcoin transaction to a Bitcoin address that does not exist? Say I arbitrarily pick an address "abcdefg" (for example's sake), which does not exist in the network; no one "owns" the address.

What happens when I attempt to make a transaction to this address?

Is the transaction rejected by the network?

If not is this transaction stored in the blockchain?
Can you send a bitcoin transaction to a bitcoin address that does not exist?


Absolutely, however, the address must be valid. 

A completely arbitrary address (or a typo) will likely not be valid because addresses contain a checksum when they are Base58 encoded. There is no way for the network to know if someone owns a key corresponding to an address unless a transaction has occurred with that address. See Bitcoin Wiki - Address and BIP 13 which describes how a Pay-to-Script-Hash address is formed, as an example:

base58-encode: [one-byte version][20-byte hash][4-byte checksum]