**Gannibal** [accept Abraham **Hannibal**]
[10] During Peter's minority, he cut a hole in the back of his throne so that this sister of his, the regent, could feed him responses while he conversed with the boyars. She was allied with the Streltsy and was locked in a convent when that group rebelled.
Answer: **Sophia** Alekseyevna
2. One of them states that if a power of a prime divides the order of a group G, then G has at least one subgroup of order equal to the power of the prime. For 10 points each:
[10] Name this set of three theorems, all dealing with subgroup orders and primes.
ANSWER: **Sylow** theorems
[10] A corollary to Sylow's first theorem is named for this French mathematician who also names an inequality with Schwarz, a formula with Binet, a convergence test, and types of sequences, among other things.
ANSWER: Augustin Louis **Cauchy**
[10] These seven groups describing different symmetries in two-dimensional art are nicknamed “hop,” “step,” “jump,” “sidle,” “spinning hop,” “spinning step,” and “spinning jump.” They describe one-directional symmetries, while the Wallpaper groups describe two-directional symmetries.
ANSWER: **Frieze** groups
3. Name these weapons from European mythologies, for 10 points each.
[10] Used to kill Ferdiad in the Cattle Raid of Cooley, this spear, made from the bones of a sea monster, was given to Cu Chulainn by Aife and accidentally killed their son Connla.
ANSWER:
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