Sage (comics) New Excalibur What was New Excalibur? For reasons yet unknown, Sage leaves Sunspot's side and travels to England to join forces with the newly reformed New Excalibur.
Is that a group?
Is New Excalibur a group?
For reasons yet unknown, Sage leaves Sunspot's side and travels to England to join forces with the newly reformed New Excalibur. She helps them on various missions such as battling the Shadow King, and returning to ancient Camelot to prevent it from being prematurely destroyed. In a more subtle use of her observation abilities, Sage is instrumental in her teammate Nocturne's recovery from a debilitating stroke.  In order to learn more about a mysterious new foe, Albion, and thwart his attempt to conquer Britain, Sage creates the new cover identity and persona of Diana Fox. This persona appears to be fully devoted to Albion and his goals. However, this bothers her teammate Wisdom as he fears she might fail the mission, similar to her time with the Hellfire Club. Her Diana Fox persona climbs up the rank of Albion's Shadow Captains as Britannia, and develops a rivalry with Lionheart. Like the rest of Albion's army, Sage is given a suit that emphasizes loyalty to him, as well as equipping her with flight and superhuman physical strength. Although she has the opportunity to kill Albion, one purpose of her infiltration, she is struck by an errant pulse from an ancient item used by Albion to extinguish power in England. Though Lionheart suspects her betrayal, Albion believes she was only trying to protect him.  Wisdom's fears are realized after this when the Diana Fox persona takes complete control and Sage becomes fully committed to Albion's side. During this time, she slays two of New Excalibur's former allies in Shadow-X, Dark Cyclops and Dark Beast. A direct confrontation with her teammates in New Excalibur helps reassert Sage's original personality. Together with New Excalibur, Shadow-X, and now Lionheart, Sage defeats Albion and his Shadow Captains. CANNOTANSWER
Together with New Excalibur, Shadow-X, and now Lionheart, Sage defeats Albion and his Shadow Captains.
In mathematics, in the field of group theory, a T-group is a group in which the property of normality is transitive, that is, every subnormal subgroup is normal. Here are some facts about T-groups:

Every simple group is a T-group.
Every quasisimple group is a T-group.
Every abelian group is a T-group.
Every Hamiltonian group is a T-group.
Every nilpotent T-group is either abelian or Hamiltonian, because in a nilpotent group, every subgroup is subnormal.
Every normal subgroup of a T-group is a T-group.
Every homomorphic image of a T-group is a T-group.
Every solvable T-group is metabelian.
The solvable T-groups were characterized by Wolfgang Gaschütz as being exactly the solvable groups G with an abelian normal Hall subgroup H of odd order such that the quotient group G/H is a Dedekind group and H is acted upon by conjugation as a group of power automorphisms by G.

A PT-group is a group in which permutability is transitive. A finite T-group is a PT-group.

References
 

Properties of groups In mathematics, a semitopological group is a topological space with a group action that is continuous with respect to each variable considered separately. It is a weakening of the concept of a topological group; all topological groups are semitopological groups but the converse does not hold.

Formal definition
A semitopological group  is a topological space that is also a group such that

is continuous with respect to both  and . (Note that a topological group is continuous with reference to both variables simultaneously, and  is also required to be continuous. Here  is viewed as a topological space with the product topology.)

Clearly, every topological group is a semitopological group. To see that the converse does not hold, consider the real line  with its usual structure as an additive abelian group. Apply the lower limit topology to  with topological basis the family . Then  is continuous, but  is not continuous at 0:  is an open neighbourhood of 0 but there is no neighbourhood of 0 continued in .

It is known that any locally compact Hausdorff semitopological group is a topological group. Other similar results are also known.

See also
Lie group
Algebraic group
Compact group
Topological ring

References

Topological groups In mathematics, in the realm of group theory, the term complemented group is used in two distinct, but similar ways.

In , a complemented group is one in which every subgroup has a group-theoretic complement.  Such groups are called completely factorizable groups in the Russian literature, following  and .

The following are equivalent for any finite group G:
 G is complemented
 G is a subgroup of a direct product of groups of square-free order (a special type of Z-group)
 G is a supersolvable group with elementary abelian Sylow subgroups (a special type of A-group), .

Later, in , a group is said to be complemented if the lattice of subgroups is a complemented lattice, that is, if for every subgroup H there is a subgroup K such that H ∩ K = 1 and ⟨H, K&hairsp;⟩ is the whole group.  Hall's definition required in addition that H and K permute, that is, that HK = { hk : h in H, k in K } form a subgroup.  Such groups are also called K-groups in the Italian and lattice theoretic literature, such as .  The Frattini subgroup of a K-group is trivial; if a group has a core-free maximal subgroup that is a K-group, then it itself is a K-group; hence subgroups of K-groups need not be K-groups, but quotient groups and direct products of K-groups are K-groups, .  In  it is shown that every finite simple group is a complemented group.  Note that in the classification of finite simple groups, K-group is more used to mean a group whose proper subgroups only have composition factors amongst the known finite simple groups.

An example of a group that is not complemented (in either sense) is the cyclic group of order p2, where p is a prime number. This group only has one nontrivial subgroup H, the cyclic group of order p, so there can be no other subgroup L to be the complement of H.

References
  
 
 
 

 

Properties of groups